Moment cinétique — Wikipédia

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aria-controls="toc-Définition_et_propriétés_générales_du_moment_cinétique-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Afficher / masquer la sous-section Définition et propriétés générales du moment cinétique</span> </button> <ul id="toc-Définition_et_propriétés_générales_du_moment_cinétique-sublist" class="vector-toc-list"> <li id="toc-Cas_d'un_point_matériel" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cas_d'un_point_matériel"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Cas d'un point matériel</span> </div> </a> <ul id="toc-Cas_d'un_point_matériel-sublist" class="vector-toc-list"> <li id="toc-Théorème_du_moment_cinétique_pour_un_point_matériel" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Théorème_du_moment_cinétique_pour_un_point_matériel"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.1</span> <span>Théorème du moment cinétique pour un point matériel</span> </div> </a> <ul id="toc-Théorème_du_moment_cinétique_pour_un_point_matériel-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exemples_d'application" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Exemples_d'application"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.2</span> <span>Exemples d'application</span> </div> </a> <ul id="toc-Exemples_d'application-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Cas_d'un_système_matériel" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cas_d'un_système_matériel"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Cas d'un système matériel</span> </div> </a> <ul id="toc-Cas_d'un_système_matériel-sublist" class="vector-toc-list"> <li id="toc-Définition" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Définition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.1</span> <span>Définition</span> </div> </a> <ul id="toc-Définition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Théorème_de_König_(Koenig)_pour_le_moment_cinétique" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Théorème_de_König_(Koenig)_pour_le_moment_cinétique"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.2</span> <span>Théorème de König (Koenig) pour le moment cinétique</span> </div> </a> <ul id="toc-Théorème_de_König_(Koenig)_pour_le_moment_cinétique-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cas_d'un_solide_:_tenseur_d'inertie" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Cas_d'un_solide_:_tenseur_d'inertie"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.3</span> <span>Cas d'un solide : tenseur d'inertie</span> </div> </a> <ul id="toc-Cas_d'un_solide_:_tenseur_d'inertie-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Moment_cinétique_et_isotropie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Moment_cinétique_et_isotropie"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Moment cinétique et isotropie</span> </div> </a> <ul id="toc-Moment_cinétique_et_isotropie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Moment_cinétique_et_mouvement_à_force_centrale" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Moment_cinétique_et_mouvement_à_force_centrale"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Moment cinétique et mouvement à force centrale</span> </div> </a> <button aria-controls="toc-Moment_cinétique_et_mouvement_à_force_centrale-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Afficher / masquer la sous-section Moment cinétique et mouvement à force centrale</span> </button> <ul id="toc-Moment_cinétique_et_mouvement_à_force_centrale-sublist" class="vector-toc-list"> <li id="toc-Cas_général" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cas_général"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Cas général</span> </div> </a> <ul id="toc-Cas_général-sublist" class="vector-toc-list"> <li id="toc-Notion_de_force_centrale" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Notion_de_force_centrale"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>Notion de force centrale</span> </div> </a> <ul id="toc-Notion_de_force_centrale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conservation_du_moment_cinétique_et_planéité_de_la_trajectoire" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Conservation_du_moment_cinétique_et_planéité_de_la_trajectoire"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.2</span> <span>Conservation du moment cinétique et planéité de la trajectoire</span> </div> </a> <ul id="toc-Conservation_du_moment_cinétique_et_planéité_de_la_trajectoire-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Loi_des_aires_et_formule_de_Binet" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Loi_des_aires_et_formule_de_Binet"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.3</span> <span>Loi des aires et formule de Binet</span> </div> </a> <ul id="toc-Loi_des_aires_et_formule_de_Binet-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Séparation_radiale-angulaire_de_l'énergie_cinétique_et_barrière_centrifuge" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Séparation_radiale-angulaire_de_l'énergie_cinétique_et_barrière_centrifuge"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.4</span> <span>Séparation radiale-angulaire de l'énergie cinétique et barrière centrifuge</span> </div> </a> <ul id="toc-Séparation_radiale-angulaire_de_l'énergie_cinétique_et_barrière_centrifuge-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Cas_où_la_force_centrale_dérive_d'une_énergie_potentielle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cas_où_la_force_centrale_dérive_d'une_énergie_potentielle"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Cas où la force centrale dérive d'une énergie potentielle</span> </div> </a> <ul id="toc-Cas_où_la_force_centrale_dérive_d'une_énergie_potentielle-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Moment_cinétique_relativiste" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Moment_cinétique_relativiste"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Moment cinétique relativiste</span> </div> </a> <ul id="toc-Moment_cinétique_relativiste-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes_et_références" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes_et_références"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Notes et références</span> </div> </a> <ul id="toc-Notes_et_références-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voir_aussi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voir_aussi"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Voir aussi</span> </div> </a> <button aria-controls="toc-Voir_aussi-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Afficher / masquer la sous-section Voir aussi</span> </button> <ul id="toc-Voir_aussi-sublist" class="vector-toc-list"> <li id="toc-Articles_connexes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Articles_connexes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Articles connexes</span> </div> </a> <ul id="toc-Articles_connexes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliographie" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bibliographie"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Bibliographie</span> </div> </a> <ul id="toc-Bibliographie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Liens_externes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Liens_externes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Liens externes</span> </div> </a> <ul id="toc-Liens_externes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Sommaire" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Basculer la table des matières" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Basculer la table des matières</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Moment cinétique</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Aller à un article dans une autre langue. Disponible en 73 langues." > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-73" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">73 langues</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Hoekmomentum" title="Hoekmomentum – afrikaans" lang="af" hreflang="af" data-title="Hoekmomentum" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B2%D8%AE%D9%85_%D8%B2%D8%A7%D9%88%D9%8A" title="زخم زاوي – arabe" lang="ar" hreflang="ar" data-title="زخم زاوي" data-language-autonym="العربية" data-language-local-name="arabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Momentu_angular" title="Momentu angular – asturien" lang="ast" hreflang="ast" data-title="Momentu angular" data-language-autonym="Asturianu" data-language-local-name="asturien" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C4%B0mpuls_momenti" title="İmpuls momenti – azerbaïdjanais" lang="az" hreflang="az" data-title="İmpuls momenti" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaïdjanais" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BC%D0%B0%D0%BD%D1%82_%D1%96%D0%BC%D0%BF%D1%83%D0%BB%D1%8C%D1%81%D1%83" title="Момант імпульсу – biélorusse" lang="be" hreflang="be" data-title="Момант імпульсу" data-language-autonym="Беларуская" data-language-local-name="biélorusse" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BC%D0%BF%D1%83%D0%BB%D1%81%D0%B0" title="Момент на импулса – bulgare" lang="bg" hreflang="bg" data-title="Момент на импулса" data-language-autonym="Български" data-language-local-name="bulgare" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A7%8C%E0%A6%A3%E0%A6%BF%E0%A6%95_%E0%A6%AD%E0%A6%B0%E0%A6%AC%E0%A7%87%E0%A6%97" title="কৌণিক ভরবেগ – bengali" lang="bn" hreflang="bn" data-title="কৌণিক ভরবেগ" data-language-autonym="বাংলা" data-language-local-name="bengali" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Ugaona_koli%C4%8Dina_kretanja" title="Ugaona količina kretanja – bosniaque" lang="bs" hreflang="bs" data-title="Ugaona količina kretanja" data-language-autonym="Bosanski" data-language-local-name="bosniaque" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Moment_angular" title="Moment angular – catalan" lang="ca" hreflang="ca" data-title="Moment angular" data-language-autonym="Català" data-language-local-name="catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Moment_hybnosti" title="Moment hybnosti – tchèque" lang="cs" hreflang="cs" data-title="Moment hybnosti" data-language-autonym="Čeština" data-language-local-name="tchèque" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%98%D0%BC%D0%BF%D1%83%D0%BB%D1%8C%D1%81_%D1%81%D0%B0%D0%BC%D0%B0%D0%BD%D1%87%C4%95" title="Импульс саманчĕ – tchouvache" lang="cv" hreflang="cv" data-title="Импульс саманчĕ" data-language-autonym="Чӑвашла" data-language-local-name="tchouvache" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Impulsmoment" title="Impulsmoment – danois" lang="da" hreflang="da" data-title="Impulsmoment" data-language-autonym="Dansk" data-language-local-name="danois" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Drehimpuls" title="Drehimpuls – allemand" lang="de" hreflang="de" data-title="Drehimpuls" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%84%CF%81%CE%BF%CF%86%CE%BF%CF%81%CE%BC%CE%AE" title="Στροφορμή – grec" lang="el" hreflang="el" data-title="Στροφορμή" data-language-autonym="Ελληνικά" data-language-local-name="grec" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Angular_momentum" title="Angular momentum – anglais" lang="en" hreflang="en" data-title="Angular momentum" data-language-autonym="English" data-language-local-name="anglais" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Angula_movokvanto" title="Angula movokvanto – espéranto" lang="eo" hreflang="eo" data-title="Angula movokvanto" data-language-autonym="Esperanto" data-language-local-name="espéranto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Momento_angular" title="Momento angular – espagnol" lang="es" hreflang="es" data-title="Momento angular" data-language-autonym="Español" data-language-local-name="espagnol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Impulsimoment" title="Impulsimoment – estonien" lang="et" hreflang="et" data-title="Impulsimoment" data-language-autonym="Eesti" data-language-local-name="estonien" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Momentu_angeluar" title="Momentu angeluar – basque" lang="eu" hreflang="eu" data-title="Momentu angeluar" data-language-autonym="Euskara" data-language-local-name="basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%DA%A9%D8%A7%D9%86%D9%87_%D8%B2%D8%A7%D9%88%DB%8C%D9%87%E2%80%8C%D8%A7%DB%8C" title="تکانه زاویه‌ای – persan" lang="fa" hreflang="fa" data-title="تکانه زاویه‌ای" data-language-autonym="فارسی" data-language-local-name="persan" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Py%C3%B6rimism%C3%A4%C3%A4r%C3%A4" title="Pyörimismäärä – finnois" lang="fi" hreflang="fi" data-title="Pyörimismäärä" data-language-autonym="Suomi" data-language-local-name="finnois" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Dreiimpuls" title="Dreiimpuls – frison septentrional" lang="frr" hreflang="frr" data-title="Dreiimpuls" data-language-autonym="Nordfriisk" data-language-local-name="frison septentrional" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/M%C3%B3iminteam_uilleach" title="Móiminteam uilleach – irlandais" lang="ga" hreflang="ga" data-title="Móiminteam uilleach" data-language-autonym="Gaeilge" data-language-local-name="irlandais" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Momento_angular" title="Momento angular – galicien" lang="gl" hreflang="gl" data-title="Momento angular" data-language-autonym="Galego" data-language-local-name="galicien" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%A0%D7%A2_%D7%96%D7%95%D7%95%D7%99%D7%AA%D7%99" title="תנע זוויתי – hébreu" lang="he" hreflang="he" data-title="תנע זוויתי" data-language-autonym="עברית" data-language-local-name="hébreu" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%95%E0%A5%8B%E0%A4%A3%E0%A5%80%E0%A4%AF_%E0%A4%B8%E0%A4%82%E0%A4%B5%E0%A5%87%E0%A4%97" title="कोणीय संवेग – hindi" lang="hi" hreflang="hi" data-title="कोणीय संवेग" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Kutna_koli%C4%8Dina_gibanja" title="Kutna količina gibanja – croate" lang="hr" hreflang="hr" data-title="Kutna količina gibanja" data-language-autonym="Hrvatski" data-language-local-name="croate" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Perd%C3%BClet" title="Perdület – hongrois" lang="hu" hreflang="hu" data-title="Perdület" data-language-autonym="Magyar" data-language-local-name="hongrois" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BB%D5%B4%D5%BA%D5%B8%D6%82%D5%AC%D5%BD%D5%AB_%D5%B4%D5%B8%D5%B4%D5%A5%D5%B6%D5%BF" title="Իմպուլսի մոմենտ – arménien" lang="hy" hreflang="hy" data-title="Իմպուլսի մոմենտ" data-language-autonym="Հայերեն" data-language-local-name="arménien" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Momentum_sudut" title="Momentum sudut – indonésien" lang="id" hreflang="id" data-title="Momentum sudut" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésien" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Hverfi%C3%BEungi" title="Hverfiþungi – islandais" lang="is" hreflang="is" data-title="Hverfiþungi" data-language-autonym="Íslenska" data-language-local-name="islandais" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Momento_angolare" title="Momento angolare – italien" lang="it" hreflang="it" data-title="Momento angolare" data-language-autonym="Italiano" data-language-local-name="italien" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%A7%92%E9%81%8B%E5%8B%95%E9%87%8F" title="角運動量 – japonais" lang="ja" hreflang="ja" data-title="角運動量" data-language-autonym="日本語" data-language-local-name="japonais" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%98%E1%83%9B%E1%83%9E%E1%83%A3%E1%83%9A%E1%83%A1%E1%83%98%E1%83%A1_%E1%83%9B%E1%83%9D%E1%83%9B%E1%83%94%E1%83%9C%E1%83%A2%E1%83%98" title="იმპულსის მომენტი – géorgien" lang="ka" hreflang="ka" data-title="იმპულსის მომენტი" data-language-autonym="ქართული" data-language-local-name="géorgien" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Imir_u%C9%A3mir" title="Imir uɣmir – kabyle" lang="kab" hreflang="kab" data-title="Imir uɣmir" data-language-autonym="Taqbaylit" data-language-local-name="kabyle" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%98%D0%BC%D0%BF%D1%83%D0%BB%D1%8C%D1%81_%D0%BC%D0%BE%D0%BC%D0%B5%D0%BD%D1%82%D1%96" title="Импульс моменті – kazakh" lang="kk" hreflang="kk" data-title="Импульс моменті" data-language-autonym="Қазақша" data-language-local-name="kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B0%81%EC%9A%B4%EB%8F%99%EB%9F%89" title="각운동량 – coréen" lang="ko" hreflang="ko" data-title="각운동량" data-language-autonym="한국어" data-language-local-name="coréen" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Judesio_kiekio_momentas" title="Judesio kiekio momentas – lituanien" lang="lt" hreflang="lt" data-title="Judesio kiekio momentas" data-language-autonym="Lietuvių" data-language-local-name="lituanien" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Impulsa_moments" title="Impulsa moments – letton" lang="lv" hreflang="lv" data-title="Impulsa moments" data-language-autonym="Latviešu" data-language-local-name="letton" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BC%D0%BF%D1%83%D0%BB%D1%81%D0%BE%D1%82" title="Момент на импулсот – macédonien" lang="mk" hreflang="mk" data-title="Момент на импулсот" data-language-autonym="Македонски" data-language-local-name="macédonien" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%95%E0%A5%8B%E0%A4%A8%E0%A5%80%E0%A4%AF_%E0%A4%B8%E0%A4%82%E0%A4%B5%E0%A5%87%E0%A4%97" title="कोनीय संवेग – marathi" lang="mr" hreflang="mr" data-title="कोनीय संवेग" data-language-autonym="मराठी" data-language-local-name="marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Momentum_sudut" title="Momentum sudut – malais" lang="ms" hreflang="ms" data-title="Momentum sudut" data-language-autonym="Bahasa Melayu" data-language-local-name="malais" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%91%E1%80%B1%E1%80%AC%E1%80%84%E1%80%B7%E1%80%BA%E1%80%95%E1%80%BC%E1%80%B1%E1%80%AC%E1%80%84%E1%80%BA%E1%80%B8%E1%80%A1%E1%80%9F%E1%80%AF%E1%80%94%E1%80%BA" title="ထောင့်ပြောင်းအဟုန် – birman" lang="my" hreflang="my" data-title="ထောင့်ပြောင်းအဟုန်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birman" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Impulsmoment" title="Impulsmoment – néerlandais" lang="nl" hreflang="nl" data-title="Impulsmoment" data-language-autonym="Nederlands" data-language-local-name="néerlandais" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Vinkelmoment" title="Vinkelmoment – norvégien nynorsk" lang="nn" hreflang="nn" data-title="Vinkelmoment" data-language-autonym="Norsk nynorsk" data-language-local-name="norvégien nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Drivmoment" title="Drivmoment – norvégien bokmål" lang="nb" hreflang="nb" data-title="Drivmoment" data-language-autonym="Norsk bokmål" data-language-local-name="norvégien bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Moment_cinetic" title="Moment cinetic – occitan" lang="oc" hreflang="oc" data-title="Moment cinetic" data-language-autonym="Occitan" data-language-local-name="occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%90%E0%A8%82%E0%A8%97%E0%A9%81%E0%A8%B2%E0%A8%B0_%E0%A8%AE%E0%A9%8B%E0%A8%AE%E0%A9%88%E0%A8%82%E0%A8%9F%E0%A8%AE" title="ਐਂਗੁਲਰ ਮੋਮੈਂਟਮ – pendjabi" lang="pa" hreflang="pa" data-title="ਐਂਗੁਲਰ ਮੋਮੈਂਟਮ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="pendjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Moment_p%C4%99du" title="Moment pędu – polonais" lang="pl" hreflang="pl" data-title="Moment pędu" data-language-autonym="Polski" data-language-local-name="polonais" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Moment_angolar" title="Moment angolar – piémontais" lang="pms" hreflang="pms" data-title="Moment angolar" data-language-autonym="Piemontèis" data-language-local-name="piémontais" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Momento_angular" title="Momento angular – portugais" lang="pt" hreflang="pt" data-title="Momento angular" data-language-autonym="Português" data-language-local-name="portugais" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Moment_cinetic" title="Moment cinetic – roumain" lang="ro" hreflang="ro" data-title="Moment cinetic" data-language-autonym="Română" data-language-local-name="roumain" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%B8%D0%BC%D0%BF%D1%83%D0%BB%D1%8C%D1%81%D0%B0" title="Момент импульса – russe" lang="ru" hreflang="ru" data-title="Момент импульса" data-language-autonym="Русский" data-language-local-name="russe" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Ugaoni_moment" title="Ugaoni moment – serbo-croate" lang="sh" hreflang="sh" data-title="Ugaoni moment" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-croate" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Angular_momentum" title="Angular momentum – Simple English" lang="en-simple" hreflang="en-simple" data-title="Angular momentum" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Moment_hybnosti" title="Moment hybnosti – slovaque" lang="sk" hreflang="sk" data-title="Moment hybnosti" data-language-autonym="Slovenčina" data-language-local-name="slovaque" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Vrtilna_koli%C4%8Dina" title="Vrtilna količina – slovène" lang="sl" hreflang="sl" data-title="Vrtilna količina" data-language-autonym="Slovenščina" data-language-local-name="slovène" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Impulsi_k%C3%ABndor" title="Impulsi këndor – albanais" lang="sq" hreflang="sq" data-title="Impulsi këndor" data-language-autonym="Shqip" data-language-local-name="albanais" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%B8%D0%BC%D0%BF%D1%83%D0%BB%D1%81%D0%B0" title="Момент импулса – serbe" lang="sr" hreflang="sr" data-title="Момент импулса" data-language-autonym="Српски / srpski" data-language-local-name="serbe" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Mom%C3%A9ntum_sudut" title="Moméntum sudut – soundanais" lang="su" hreflang="su" data-title="Moméntum sudut" data-language-autonym="Sunda" data-language-local-name="soundanais" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/R%C3%B6relsem%C3%A4ngdsmoment" title="Rörelsemängdsmoment – suédois" lang="sv" hreflang="sv" data-title="Rörelsemängdsmoment" data-language-autonym="Svenska" data-language-local-name="suédois" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AE%B3%E0%AF%88%E0%AE%B5%E0%AF%81%E0%AE%A8%E0%AF%8D%E0%AE%A4%E0%AE%AE%E0%AF%8D" title="வளைவுந்தம் – tamoul" lang="ta" hreflang="ta" data-title="வளைவுந்தம்" data-language-autonym="தமிழ்" data-language-local-name="tamoul" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%95%E0%B1%8B%E0%B0%A3%E0%B1%80%E0%B0%AF_%E0%B0%A6%E0%B1%8D%E0%B0%B0%E0%B0%B5%E0%B1%8D%E0%B0%AF%E0%B0%B5%E0%B1%87%E0%B0%97%E0%B0%82" title="కోణీయ ద్రవ్యవేగం – télougou" lang="te" hreflang="te" data-title="కోణీయ ద్రవ్యవేగం" data-language-autonym="తెలుగు" data-language-local-name="télougou" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%82%E0%B8%A1%E0%B9%80%E0%B8%A1%E0%B8%99%E0%B8%95%E0%B8%B1%E0%B8%A1%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B8%A1%E0%B8%B8%E0%B8%A1" title="โมเมนตัมเชิงมุม – thaï" lang="th" hreflang="th" data-title="โมเมนตัมเชิงมุม" data-language-autonym="ไทย" data-language-local-name="thaï" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/A%C3%A7%C4%B1sal_momentum" title="Açısal momentum – turc" lang="tr" hreflang="tr" data-title="Açısal momentum" data-language-autonym="Türkçe" data-language-local-name="turc" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%C4%B0mpuls_moment%C4%B1" title="İmpuls momentı – tatar" lang="tt" hreflang="tt" data-title="İmpuls momentı" data-language-autonym="Татарча / tatarça" data-language-local-name="tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D1%96%D0%BC%D0%BF%D1%83%D0%BB%D1%8C%D1%81%D1%83" title="Момент імпульсу – ukrainien" lang="uk" hreflang="uk" data-title="Момент імпульсу" data-language-autonym="Українська" data-language-local-name="ukrainien" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B2%D8%A7%D9%88%DB%8C%D8%A7%D8%A6%DB%8C_%D9%85%D8%B9%DB%8C%D8%A7%D8%B1_%D8%AD%D8%B1%DA%A9%D8%AA" title="زاویائی معیار حرکت – ourdou" lang="ur" hreflang="ur" data-title="زاویائی معیار حرکت" data-language-autonym="اردو" data-language-local-name="ourdou" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/M%C3%B4_men_%C4%91%E1%BB%99ng_l%C6%B0%E1%BB%A3ng" title="Mô men động lượng – vietnamien" lang="vi" hreflang="vi" data-title="Mô men động lượng" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamien" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E8%A7%92%E5%8A%A8%E9%87%8F" title="角动量 – wu" lang="wuu" hreflang="wuu" data-title="角动量" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%A7%92%E5%8A%A8%E9%87%8F" title="角动量 – chinois" lang="zh" hreflang="zh" data-title="角动量" data-language-autonym="中文" data-language-local-name="chinois" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a 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<button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">déplacer vers la barre latérale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">masquer</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Un article de Wikipédia, l'encyclopédie libre.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="fr" dir="ltr"><div class="bandeau-container metadata homonymie hatnote"><div class="bandeau-cell bandeau-icone" style="display:table-cell;padding-right:0.5em"><span class="noviewer" typeof="mw:File"><a href="/wiki/Aide:Homonymie" title="Aide:Homonymie"><img alt="Page d’aide sur l’homonymie" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/20px-Logo_disambig.svg.png" decoding="async" width="20" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/30px-Logo_disambig.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/40px-Logo_disambig.svg.png 2x" data-file-width="512" data-file-height="375" /></a></span></div><div class="bandeau-cell" style="display:table-cell;padding-right:0.5em"> <p>Pour les articles homonymes, voir <a href="/wiki/Moment" class="mw-disambig" title="Moment">Moment</a>. </p> </div></div> <div class="infobox_v3 infobox infobox--frwiki noarchive"> <div class="entete" style=""> <div>Moment cinétique</div> </div> <div class="images"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/Fichier:Gyroskop.jpg" class="mw-file-description"><img alt="Description de cette image, également commentée ci-après" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Gyroskop.jpg/220px-Gyroskop.jpg" decoding="async" width="220" height="346" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/b/b9/Gyroskop.jpg 1.5x" data-file-width="282" data-file-height="444" /></a></span> </div> <div class="legend">Un <a href="/wiki/Gyroscope" title="Gyroscope">gyroscope</a> tournant sur un clou.</div><table><caption class="hidden" style="">Données clés</caption> <tbody><tr> <th scope="row"><a href="/wiki/Unit%C3%A9s_de_base_du_Syst%C3%A8me_international" title="Unités de base du Système international">Unités SI</a></th> <td> <abbr class="abbr" title="kilogramme mètre carré radian par seconde">kg m<sup>2</sup> rad/s</abbr></td> </tr> <tr> <th scope="row"><a href="/wiki/Dimension_(physique)" title="Dimension (physique)">Dimension</a></th> <td> <span class="nowrap"><a href="/wiki/Masse" title="Masse">M</a>·<a href="/wiki/Longueur" title="Longueur">L</a><sup> 2</sup>·<a href="/wiki/Temps_(physique)" title="Temps (physique)">T</a><sup> −1</sup></span></td> </tr> <tr> <th scope="row">Nature</th> <td> Grandeur <a href="/wiki/Pseudovecteur" title="Pseudovecteur">vectorielle (pseudovecteur)</a> conservative <a href="/wiki/Grandeur_extensive" class="mw-redirect" title="Grandeur extensive">extensive</a></td> </tr> <tr> <th scope="row">Symbole usuel</th> <td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {L}}_{\mathrm {O} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {L}}_{\mathrm {O} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9477090acef4968b82a9a4ab592fd12ddfb0863" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.094ex; height:3.176ex;" alt="{\displaystyle {\vec {L}}_{\mathrm {O} }}"></span> ou <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\sigma }}_{\mathrm {O} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\sigma }}_{\mathrm {O} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef9dafb4f6898f08f72dc3ad03cbe6912341d8c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.84ex; height:2.676ex;" alt="{\displaystyle {\vec {\sigma }}_{\mathrm {O} }}"></span></td> </tr> <tr> <th scope="row">Lien à d'autres grandeurs</th> <td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {L}}_{\mathrm {O} }={\overrightarrow {\mathrm {OM} }}\wedge {\vec {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">M</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {L}}_{\mathrm {O} }={\overrightarrow {\mathrm {OM} }}\wedge {\vec {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9b1197929a6b0f12fec7b6e757ef7f05664806f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.398ex; width:14.169ex; height:4.176ex;" alt="{\displaystyle {\vec {L}}_{\mathrm {O} }={\overrightarrow {\mathrm {OM} }}\wedge {\vec {p}}}"></span></td> </tr> <tr> <th scope="row"><a href="/wiki/Variables_conjugu%C3%A9es_(thermodynamique)" title="Variables conjuguées (thermodynamique)">Conjuguée</a></th> <td> <a href="/wiki/Vitesse_de_rotation" class="mw-redirect" title="Vitesse de rotation">Vitesse de rotation</a></td> </tr> <tr> <th scope="row"><a href="/wiki/Principe_de_compl%C3%A9mentarit%C3%A9" title="Principe de complémentarité">Grandeur duale</a></th> <td> <a href="/wiki/Angle_plan" title="Angle plan">Angle plan</a></td> </tr> </tbody></table><p class="navbar bordered noprint" style=""><span class="plainlinks navigation-not-searchable"><a class="external text" href="https://fr.wikipedia.org/w/index.php?title=Moment_cin%C3%A9tique&action=edit">modifier</a></span> <span typeof="mw:File"><a href="/wiki/Mod%C3%A8le:Infobox_Grandeur_physique" title="Consultez la documentation du modèle"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Info_Simple.svg/12px-Info_Simple.svg.png" decoding="async" width="12" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Info_Simple.svg/18px-Info_Simple.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/38/Info_Simple.svg/24px-Info_Simple.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></p></div> <p>En <a href="/wiki/M%C3%A9canique_newtonienne" title="Mécanique newtonienne">mécanique classique</a>, le <b>moment cinétique</b> (ou <b>moment angulaire</b> par <a href="/wiki/Anglicisme" title="Anglicisme">anglicisme</a>) d'un <a href="/wiki/Point_mat%C3%A9riel" title="Point matériel">point matériel</a> M par rapport à un point O est le <a href="/wiki/Moment_d%27un_vecteur" title="Moment d'un vecteur">moment</a> de la <a href="/wiki/Quantit%C3%A9_de_mouvement" title="Quantité de mouvement">quantité de mouvement</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84fee53c81592db54e0fe6c6f9eba002bb1dc74b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}"></span> par rapport au point O, c'est-à-dire le <a href="/wiki/Produit_vectoriel#Définition" title="Produit vectoriel">produit vectoriel</a> : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {L}}_{\mathrm {O} }={\overrightarrow {\mathrm {OM} }}\wedge {\vec {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">M</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {L}}_{\mathrm {O} }={\overrightarrow {\mathrm {OM} }}\wedge {\vec {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9b1197929a6b0f12fec7b6e757ef7f05664806f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.398ex; width:14.169ex; height:4.176ex;" alt="{\displaystyle {\vec {L}}_{\mathrm {O} }={\overrightarrow {\mathrm {OM} }}\wedge {\vec {p}}}"></span>.</dd></dl> <p>Le moment cinétique d'un <a href="/wiki/Syst%C3%A8me_physique" title="Système physique">système matériel</a> est la somme des moments cinétiques (par rapport au même point O) des points matériels constituant le système<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup> : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {L}}_{\mathrm {O} }=\sum _{i}{\overrightarrow {\mathrm {OM} _{i}}}\wedge {\vec {p}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>∧</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {L}}_{\mathrm {O} }=\sum _{i}{\overrightarrow {\mathrm {OM} _{i}}}\wedge {\vec {p}}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dd87df9e9bbd46026b96d508a8757210019e2e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-top: -0.398ex; width:19.51ex; height:6.509ex;" alt="{\displaystyle {\vec {L}}_{\mathrm {O} }=\sum _{i}{\overrightarrow {\mathrm {OM} _{i}}}\wedge {\vec {p}}_{i}}"></span>.</dd></dl> <p>Cette grandeur, considérée dans un <a href="/wiki/R%C3%A9f%C3%A9rentiel_galil%C3%A9en" title="Référentiel galiléen">référentiel galiléen</a>, dépend du choix de l'origine O ; par suite, il n'est pas possible de combiner en général des moments angulaires ayant des origines différentes. Son unité est le <a href="/wiki/Kilogramme" title="Kilogramme">kilogramme</a> <a href="/wiki/M%C3%A8tre_carr%C3%A9" title="Mètre carré">mètre carré</a> <a href="/wiki/Radian" title="Radian">radian</a> par <a href="/wiki/Seconde_(temps)" title="Seconde (temps)">seconde</a> (<abbr class="abbr" title="kilogramme mètre carré radian par seconde">kg m<sup>2</sup> rad/s</abbr>). Par ailleurs, il s'agit d'un <a href="/wiki/Champ_%C3%A9quiprojectif" title="Champ équiprojectif">champ équiprojectif</a>, donc un <a href="/wiki/Torseur_cin%C3%A9tique" title="Torseur cinétique">torseur</a>. </p><p>Le moment cinétique joue, dans le cas d'une rotation, un <a href="/wiki/Analogie_entre_rotation_et_translation" title="Analogie entre rotation et translation">rôle analogue</a> à celui de la <a href="/wiki/Quantit%C3%A9_de_mouvement" title="Quantité de mouvement">quantité de mouvement</a> pour une translation : si la conservation de la quantité de mouvement pour un <a href="/wiki/Syst%C3%A8me_isol%C3%A9" title="Système isolé">système isolé</a> est liée à l'invariance par translation dans l'espace (propriété d'homogénéité de l'espace), la <a href="/wiki/Conservation_du_moment_cin%C3%A9tique" title="Conservation du moment cinétique">conservation du moment cinétique</a> est liée à l'isotropie de l'espace. Le lien entre moment angulaire et rotation est encore plus net en <a href="/wiki/M%C3%A9canique_analytique" title="Mécanique analytique">mécanique analytique</a> et surtout en <a href="/wiki/M%C3%A9canique_quantique" title="Mécanique quantique">mécanique quantique</a> (cf. <a href="/wiki/Moment_cin%C3%A9tique_(m%C3%A9canique_quantique)" title="Moment cinétique (mécanique quantique)">moment cinétique en mécanique quantique</a>) où ce concept est enrichi, avec l'apparition d'un moment cinétique sans équivalent classique (le <a href="/wiki/Spin" title="Spin">spin</a>). </p><p>Pour un point matériel, la variation temporelle du moment cinétique est donnée par la somme des <a href="/wiki/Moment_d%27une_force" title="Moment d'une force">moments des forces</a> appliquées à ce point. Ce résultat, qui peut se généraliser à un système de points, constitue le <a href="/wiki/Th%C3%A9or%C3%A8me_du_moment_cin%C3%A9tique" title="Théorème du moment cinétique">théorème du moment cinétique</a> et est l'analogue de la <a href="/wiki/Principe_fondamental_de_la_dynamique" title="Principe fondamental de la dynamique">relation fondamentale de la dynamique</a>, qui lie variation temporelle de la quantité de mouvement et somme des forces appliquées. </p><p>C'est un <a href="/wiki/Op%C3%A9rateur_(physique)" title="Opérateur (physique)">opérateur</a> en <a href="/wiki/M%C3%A9canique_quantique" title="Mécanique quantique">mécanique quantique</a>, et un <a href="/wiki/Tenseur" title="Tenseur">tenseur</a> en <a href="/wiki/Relativit%C3%A9_restreinte" title="Relativité restreinte">relativité restreinte</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Définition_et_propriétés_générales_du_moment_cinétique"><span id="D.C3.A9finition_et_propri.C3.A9t.C3.A9s_g.C3.A9n.C3.A9rales_du_moment_cin.C3.A9tique"></span>Définition et propriétés générales du moment cinétique</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moment_cin%C3%A9tique&veaction=edit&section=1" title="Modifier la section : Définition et propriétés générales du moment cinétique" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Moment_cin%C3%A9tique&action=edit&section=1" title="Modifier le code source de la section : Définition et propriétés générales du moment cinétique"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pour définir simplement la notion de moment cinétique, il est utile de prendre d'abord le cas simple d'un <i><a href="/wiki/Point_mat%C3%A9riel" title="Point matériel">point matériel</a></i> (ou <a href="/wiki/Corps_ponctuel" class="mw-redirect" title="Corps ponctuel">corps ponctuel</a>), qui correspond à une idéalisation où les dimensions d'un système sont considérées comme petites devant les distances caractéristiques du mouvement étudié (distance parcourue, rayon d'une orbite…). Le système est alors modélisé par un simple point géométrique (noté <i>M</i>) auquel est associée sa masse <i>m</i>. Il est ensuite possible de généraliser par additivité la notion de moment cinétique à un système quelconque, considéré comme un ensemble de points matériels. </p> <div class="mw-heading mw-heading3"><h3 id="Cas_d'un_point_matériel"><span id="Cas_d.27un_point_mat.C3.A9riel"></span>Cas d'un point matériel</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moment_cin%C3%A9tique&veaction=edit&section=2" title="Modifier la section : Cas d'un point matériel" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Moment_cin%C3%A9tique&action=edit&section=2" title="Modifier le code source de la section : Cas d'un point matériel"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pour un point matériel M en mouvement par rapport à un <a href="/wiki/R%C3%A9f%C3%A9rentiel_(physique)" title="Référentiel (physique)">référentiel</a> donné, de vecteur position <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}={\overrightarrow {\mathrm {OM} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">M</mi> </mrow> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}={\overrightarrow {\mathrm {OM} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e4cd1a56f1cd6efafc6af4a48775f2f838cbf62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-top: -0.398ex; width:8.391ex; height:3.843ex;" alt="{\displaystyle {\vec {r}}={\overrightarrow {\mathrm {OM} }}}"></span>, le moment cinétique (ou angulaire) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {L_{\mathrm {O} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {L_{\mathrm {O} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0edc84312e83478abf89fc597cd9ab901baf07b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.449ex; width:3.224ex; height:4.176ex;" alt="{\displaystyle {\overrightarrow {L_{\mathrm {O} }}}}"></span> par rapport à un point O choisi comme origine est défini ainsi<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup><sup class="reference cite_virgule">,</sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite_crochet">[</span>3<span class="cite_crochet">]</span></a></sup><sup class="reference cite_virgule">,</sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite_crochet">[</span>4<span class="cite_crochet">]</span></a></sup> : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {L_{\mathrm {O} }}}={\overrightarrow {\mathrm {OM} }}\wedge {\vec {p}}={\vec {r}}\wedge {\vec {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">M</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {L_{\mathrm {O} }}}={\overrightarrow {\mathrm {OM} }}\wedge {\vec {p}}={\vec {r}}\wedge {\vec {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efbdbb86c34d62b39b7214c3da2b4b87e8ed6d93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.398ex; width:22.528ex; height:4.176ex;" alt="{\displaystyle {\overrightarrow {L_{\mathrm {O} }}}={\overrightarrow {\mathrm {OM} }}\wedge {\vec {p}}={\vec {r}}\wedge {\vec {p}}}"></span>,</dd></dl> <p>où <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}=m\,{\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>m</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p}}=m\,{\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6fe6edcfba1d8589fd1b5962d3273052c1b10f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.116ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}=m\,{\vec {v}}}"></span> est la <a href="/wiki/Quantit%C3%A9_de_mouvement" title="Quantité de mouvement">quantité de mouvement</a> de la particule. Le moment cinétique est donc le <a href="/wiki/Moment_d%27une_force" title="Moment d'une force">moment</a> de cette dernière par rapport à O. Il dépend du point O ainsi que du référentiel d'étude. </p><p>Si le mouvement du point matériel par rapport au référentiel considéré est rectiligne et que le point O se trouve sur la trajectoire, les vecteurs position et quantité de mouvement seront colinéaires, et le moment cinétique sera nul. En revanche, pour un point O en dehors de la trajectoire, du point de vue duquel la direction de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84fee53c81592db54e0fe6c6f9eba002bb1dc74b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}"></span> « tourne » par rapport à celle de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aec3c9ce13b53e9e24c98e7cce4212627884c91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}"></span>, cela ne sera plus vrai et le moment cinétique ne sera plus nul. Intuitivement, ceci permet de voir que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {L_{\mathrm {O} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {L_{\mathrm {O} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0edc84312e83478abf89fc597cd9ab901baf07b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.449ex; width:3.224ex; height:4.176ex;" alt="{\displaystyle {\overrightarrow {L_{\mathrm {O} }}}}"></span> est lié d'une certaine manière à la « rotation » du point matériel M autour de l'origine O. </p><p>Il est possible de traduire cette vision intuitive de façon générale et quantitative en établissant la relation entre sa variation temporelle (dérivée) du moment cinétique et la somme des moments des <a href="/wiki/Forces_int%C3%A9rieures_et_forces_ext%C3%A9rieures" title="Forces intérieures et forces extérieures">forces extérieures</a> appliquées au système : c'est le <i>théorème du moment cinétique</i>. </p> <div class="mw-heading mw-heading4"><h4 id="Théorème_du_moment_cinétique_pour_un_point_matériel"><span id="Th.C3.A9or.C3.A8me_du_moment_cin.C3.A9tique_pour_un_point_mat.C3.A9riel"></span>Théorème du moment cinétique pour un point matériel</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moment_cin%C3%A9tique&veaction=edit&section=3" title="Modifier la section : Théorème du moment cinétique pour un point matériel" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Moment_cin%C3%A9tique&action=edit&section=3" title="Modifier le code source de la section : Théorème du moment cinétique pour un point matériel"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La dérivation membre à membre de l'expression du moment angulaire, en notant que O est supposé fixe dans le référentiel d'étude (<i>R</i>), permet d'obtenir : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} {\overrightarrow {L_{\mathrm {O} }}}}{\mathrm {d} t}}={\frac {\mathrm {d} {\vec {r}}}{\mathrm {d} t}}\wedge {\vec {p}}+{\vec {r}}\wedge {\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}}={\vec {r}}\wedge {\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} {\overrightarrow {L_{\mathrm {O} }}}}{\mathrm {d} t}}={\frac {\mathrm {d} {\vec {r}}}{\mathrm {d} t}}\wedge {\vec {p}}+{\vec {r}}\wedge {\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}}={\vec {r}}\wedge {\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/565b2ca18e815a401cf3b0dc09bfc41017e02550" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-top: -0.344ex; width:36.168ex; height:7.009ex;" alt="{\displaystyle {\frac {\mathrm {d} {\overrightarrow {L_{\mathrm {O} }}}}{\mathrm {d} t}}={\frac {\mathrm {d} {\vec {r}}}{\mathrm {d} t}}\wedge {\vec {p}}+{\vec {r}}\wedge {\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}}={\vec {r}}\wedge {\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}}}"></span>,</dd></dl> <p>puisque <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} {\vec {r}}}{\mathrm {d} t}}\;(={\vec {v}})\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} {\vec {r}}}{\mathrm {d} t}}\;(={\vec {v}})\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20c80b025f3f5725c41d72e366c1c5941ba331eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.822ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} {\vec {r}}}{\mathrm {d} t}}\;(={\vec {v}})\,}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,{\vec {p}}\;(=m\,{\vec {v}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mo>=</mo> <mi>m</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,{\vec {p}}\;(=m\,{\vec {v}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181952927a8733788ce631b61c1c8e3fde0da0f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.223ex; height:2.843ex;" alt="{\displaystyle \,{\vec {p}}\;(=m\,{\vec {v}})}"></span> sont colinéaires. </p><p>Pour un point matériel, la <a href="/wiki/Lois_du_mouvement_de_Newton#Deuxième_loi_de_Newton_ou_principe_fondamental_de_la_dynamique" title="Lois du mouvement de Newton">relation fondamentale de la dynamique</a> s'écrit : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}}=\sum _{i}{\overrightarrow {F_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}}=\sum _{i}{\overrightarrow {F_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8749fe76ff146743f1a50f0a758f5dbb82710cfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.618ex; height:6.676ex;" alt="{\displaystyle {\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}}=\sum _{i}{\overrightarrow {F_{i}}}}"></span>.</dd></dl> <p>Le membre de droite de l'équation précédente représente la somme des forces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {F_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {F_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8d366d2e537328514aa06633eeefb5531705041" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.324ex; height:4.009ex;" alt="{\displaystyle {\overrightarrow {F_{i}}}}"></span> (réelles ou d'<a href="/wiki/Force_d%27inertie" title="Force d'inertie">inertie</a>) exercées sur le corps. </p><p>En tenant compte de ceci dans l'expression de la dérivée du moment cinétique, il vient l'équation suivante, dite <b>théorème du moment cinétique</b> : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} {\overrightarrow {L_{\mathrm {O} }}}}{\mathrm {d} t}}={\vec {r}}\wedge \sum _{i}{\overrightarrow {F_{i}}}=\sum _{i}{\overrightarrow {{\mathcal {M}}_{\mathrm {O} }}}\left({\overrightarrow {F_{i}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∧</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} {\overrightarrow {L_{\mathrm {O} }}}}{\mathrm {d} t}}={\vec {r}}\wedge \sum _{i}{\overrightarrow {F_{i}}}=\sum _{i}{\overrightarrow {{\mathcal {M}}_{\mathrm {O} }}}\left({\overrightarrow {F_{i}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef0d4870e0e3d03903b9a31792714921e0beaee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-top: -0.344ex; width:35.726ex; height:8.009ex;" alt="{\displaystyle {\frac {\mathrm {d} {\overrightarrow {L_{\mathrm {O} }}}}{\mathrm {d} t}}={\vec {r}}\wedge \sum _{i}{\overrightarrow {F_{i}}}=\sum _{i}{\overrightarrow {{\mathcal {M}}_{\mathrm {O} }}}\left({\overrightarrow {F_{i}}}\right)}"></span>,</dd></dl> <p>où <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {{\mathcal {M}}_{\mathrm {O} }}}\left({\overrightarrow {F_{i}}}\right)={\vec {r}}\wedge {\overrightarrow {F_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {{\mathcal {M}}_{\mathrm {O} }}}\left({\overrightarrow {F_{i}}}\right)={\vec {r}}\wedge {\overrightarrow {F_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6a723764338fc221c5ff21114b64242a59ce9ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.791ex; height:6.176ex;" alt="{\displaystyle {\overrightarrow {{\mathcal {M}}_{\mathrm {O} }}}\left({\overrightarrow {F_{i}}}\right)={\vec {r}}\wedge {\overrightarrow {F_{i}}}}"></span> est le <b><a href="/wiki/Moment_d%27une_force" title="Moment d'une force">moment</a> de la force</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {F_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {F_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8d366d2e537328514aa06633eeefb5531705041" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.324ex; height:4.009ex;" alt="{\displaystyle {\overrightarrow {F_{i}}}}"></span> par rapport au point O supposé fixe dans le référentiel<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite_crochet">[</span>5<span class="cite_crochet">]</span></a></sup>. Cette grandeur (appelée en anglais <i>torque</i>) correspond donc à la variation du moment cinétique en O qu'engendre l'action de la force <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {F_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {F_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8d366d2e537328514aa06633eeefb5531705041" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.324ex; height:4.009ex;" alt="{\displaystyle {\overrightarrow {F_{i}}}}"></span>. </p><p>Il en découle que si le moment résultant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}{\overrightarrow {{\mathcal {M}}_{\mathrm {O} }}}\left({\overrightarrow {F_{i}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}{\overrightarrow {{\mathcal {M}}_{\mathrm {O} }}}\left({\overrightarrow {F_{i}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e65ec13c087c92ab3dec528582f3dcdd361bc0df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.306ex; height:6.676ex;" alt="{\displaystyle \sum _{i}{\overrightarrow {{\mathcal {M}}_{\mathrm {O} }}}\left({\overrightarrow {F_{i}}}\right)}"></span> est nul, alors le moment cinétique <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {L_{\mathrm {O} }}}={\overrightarrow {\text{Cte}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mtext>Cte</mtext> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {L_{\mathrm {O} }}}={\overrightarrow {\text{Cte}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bd77100ec5ba5e506369a4ac0abebe46451a973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.398ex; width:10.067ex; height:4.176ex;" alt="{\displaystyle {\overrightarrow {L_{\mathrm {O} }}}={\overrightarrow {\text{Cte}}}}"></span> est une <i>intégrale première</i> du mouvement. </p><p><span style="text-decoration: underline;"><i>Interprétation physique</i></span> : le théorème du moment cinétique est similaire dans sa forme à la <a href="/wiki/Principe_fondamental_de_la_dynamique" title="Principe fondamental de la dynamique">relation fondamentale de la dynamique</a>. Si cette dernière relie forces appliquées au point matériel et variation de sa quantité de mouvement, le théorème du moment cinétique relie la somme des moments de ces forces par rapport à un point donné et la variation du moment cinétique par rapport à ce même point. Or le <a href="/wiki/Moment_d%27une_force" title="Moment d'une force">moment d'une force</a> par rapport à un point traduit en quelque sorte la « propension » de cette force à faire « tourner » le système autour de ce point<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite_crochet">[</span>6<span class="cite_crochet">]</span></a></sup>. Intuitivement, le théorème du moment cinétique est une sorte d'équivalent de la <a href="/wiki/Principe_fondamental_de_la_dynamique" title="Principe fondamental de la dynamique">relation fondamentale de la dynamique</a> pour ce qui est de la rotation du point M par rapport à O. Toutefois, c'est dans le formalisme de la <a href="/wiki/M%C3%A9canique_analytique" title="Mécanique analytique">mécanique analytique</a> que la relation étroite entre moment cinétique et rotations spatiales devient nettement plus claire. </p> <div class="mw-heading mw-heading4"><h4 id="Exemples_d'application"><span id="Exemples_d.27application"></span>Exemples d'application</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moment_cin%C3%A9tique&veaction=edit&section=4" title="Modifier la section : Exemples d'application" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Moment_cin%C3%A9tique&action=edit&section=4" title="Modifier le code source de la section : Exemples d'application"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div><p> Un exemple simple est celui d'une particule décrivant un cercle de centre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"></span> et de rayon <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {L_{O}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {L_{O}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a822549157837b2865e63c221b76ca5fdaa8202f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.069ex; height:4.009ex;" alt="{\displaystyle {\vec {L_{O}}}}"></span> est dirigé selon l'axe du disque et vaut <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {L_{O}}}={\vec {k}}\cdot mvr}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>m</mi> <mi>v</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {L_{O}}}={\vec {k}}\cdot mvr}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38db0feb3d114bcbdaed34878d1e03dc7608834d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.274ex; height:4.009ex;" alt="{\displaystyle {\vec {L_{O}}}={\vec {k}}\cdot mvr}"></span>. Le sens <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ccd4b98d198d6538010ae815ee1199baabd3493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.843ex;" alt="{\displaystyle {\vec {k}}}"></span> du vecteur moment cinétique ne recouvre pas une réalité physique mais est une convention ; c'est un <a href="/wiki/Pseudovecteur" title="Pseudovecteur">vecteur axial</a>.</p><figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/Fichier:Torque_animation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/09/Torque_animation.gif" decoding="async" width="220" height="154" class="mw-file-element" data-file-width="220" data-file-height="154" /></a><figcaption>Animation montrant la relation entre la force (<b>F</b>), son moment par rapport à l'origine (<b>τ</b>), le moment cinétique (<b>L</b>) relatif à cette même origine, la quantité de mouvement (<b>p</b>), pour un mouvement de rotation autour d'un axe.</figcaption></figure> <p>La figure ci-contre permet de préciser les relations entre les diverses <a href="/wiki/Grandeur_physique" title="Grandeur physique">grandeurs physiques</a>. </p><p>Par analogie avec la quantité de mouvement, le moment cinétique permet de définir l'analogue de la masse : le <a href="/wiki/Moment_d%27inertie" title="Moment d'inertie">moment d'inertie</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span>. En effet, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {L_{\mathrm {O} }}}={\vec {r}}\wedge {\vec {p}}=m{\vec {r}}\wedge {\vec {v}}=mr^{2}{\dot {\theta }}{\vec {k}}=I{\dot {\theta }}{\vec {k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>m</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {L_{\mathrm {O} }}}={\vec {r}}\wedge {\vec {p}}=m{\vec {r}}\wedge {\vec {v}}=mr^{2}{\dot {\theta }}{\vec {k}}=I{\dot {\theta }}{\vec {k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d533d75f45c042c98a2402e81ebd05c13c112ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.449ex; width:38.219ex; height:4.176ex;" alt="{\displaystyle {\overrightarrow {L_{\mathrm {O} }}}={\vec {r}}\wedge {\vec {p}}=m{\vec {r}}\wedge {\vec {v}}=mr^{2}{\dot {\theta }}{\vec {k}}=I{\dot {\theta }}{\vec {k}}}"></span></dd></dl> <p>(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f78124b2aea53527d3e053cbbdd9c7ded2c8f05f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.356ex; height:2.843ex;" alt="{\displaystyle {\dot {\theta }}}"></span> étant la <a href="/wiki/Vitesse_angulaire" title="Vitesse angulaire">vitesse angulaire</a> du point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span>), et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=mr^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mi>m</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=mr^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc416bcccf5f63d22af13befd74e5c653ab82f1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.413ex; height:2.676ex;" alt="{\displaystyle I=mr^{2}}"></span>. </p><p>En faisant correspondre à la vitesse angulaire <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f78124b2aea53527d3e053cbbdd9c7ded2c8f05f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.356ex; height:2.843ex;" alt="{\displaystyle {\dot {\theta }}}"></span> du point matériel le <a href="/wiki/Pseudovecteur" title="Pseudovecteur">vecteur axial</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\omega }}={\dot {\theta }}\,{\vec {k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\omega }}={\dot {\theta }}\,{\vec {k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2e800b9947396622bd1346ec0f220bfdebeb762" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.499ex; height:2.843ex;" alt="{\displaystyle {\vec {\omega }}={\dot {\theta }}\,{\vec {k}}}"></span>, dit vecteur rotation, le moment cinétique s'écrit finalement dans ce cas : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {L_{\mathrm {O} }}}=I\,{\vec {\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>=</mo> <mi>I</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {L_{\mathrm {O} }}}=I\,{\vec {\omega }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ea578a1a96a9251d0c1c28ac77f283137a86270" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.449ex; width:9.327ex; height:4.176ex;" alt="{\displaystyle {\overrightarrow {L_{\mathrm {O} }}}=I\,{\vec {\omega }}}"></span>.</dd></dl> <p>Dans le cas d'un solide, le moment cinétique et le vecteur rotation instantanée ne sont en général pas colinéaires, la relation entre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {L_{\mathrm {O} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {L_{\mathrm {O} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0edc84312e83478abf89fc597cd9ab901baf07b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.449ex; width:3.224ex; height:4.176ex;" alt="{\displaystyle {\overrightarrow {L_{\mathrm {O} }}}}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\omega }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e066a68ceb355e3314fb2b97f1c0c421ca6074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:2.343ex;" alt="{\displaystyle {\vec {\omega }}}"></span> fait alors intervenir un tenseur, dit d'inertie, généralisant la notion précédente. </p> <div class="mw-heading mw-heading3"><h3 id="Cas_d'un_système_matériel"><span id="Cas_d.27un_syst.C3.A8me_mat.C3.A9riel"></span>Cas d'un système matériel</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moment_cin%C3%A9tique&veaction=edit&section=5" title="Modifier la section : Cas d'un système matériel" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Moment_cin%C3%A9tique&action=edit&section=5" title="Modifier le code source de la section : Cas d'un système matériel"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Définition"><span id="D.C3.A9finition"></span>Définition</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moment_cin%C3%A9tique&veaction=edit&section=6" title="Modifier la section : Définition" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Moment_cin%C3%A9tique&action=edit&section=6" title="Modifier le code source de la section : Définition"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La notion de moment cinétique se généralise sans difficulté par additivité à un système matériel, c'est-à-dire à un corps que l'on ne peut pas assimiler à un simple point géométrique. Le moment angulaire total est obtenu en additionnant ou <a href="/wiki/Int%C3%A9gration_(math%C3%A9matiques)" title="Intégration (mathématiques)">intégrant</a> le moment angulaire de chacun de ses constituants. Il est également possible de se placer dans la limite des <a href="/wiki/M%C3%A9canique_des_milieux_continus" title="Mécanique des milieux continus">milieux continus</a> pour décrire certains systèmes mécaniques (<a href="/wiki/Mod%C3%A8le_du_solide_(m%C3%A9canique)" title="Modèle du solide (mécanique)">solides</a>, notamment). </p><p>Suivant que l'on adopte un modèle discret ou continu, le moment cinétique du système (<i>S</i>) par rapport à un point <i>O</i> s'écrit : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {L_{\mathrm {O} }}}=\sum _{i}{\overrightarrow {\mathrm {OM} _{i}}}\wedge {\vec {p_{i}}}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {L_{\mathrm {O} }}}=\sum _{i}{\overrightarrow {\mathrm {OM} _{i}}}\wedge {\vec {p_{i}}}\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66bb9e07e666d08071248defda80b2f834b4ab6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-top: -0.398ex; width:22.162ex; height:6.509ex;" alt="{\displaystyle {\overrightarrow {L_{\mathrm {O} }}}=\sum _{i}{\overrightarrow {\mathrm {OM} _{i}}}\wedge {\vec {p_{i}}}\quad }"></span> ou <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad {\overrightarrow {L_{\mathrm {O} }}}=\int _{(S)}{\overrightarrow {\mathrm {OM} }}\wedge \rho (\mathrm {M} )\,{\vec {v_{\mathrm {M} }}}\,\mathrm {d} \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">M</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>∧</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad {\overrightarrow {L_{\mathrm {O} }}}=\int _{(S)}{\overrightarrow {\mathrm {OM} }}\wedge \rho (\mathrm {M} )\,{\vec {v_{\mathrm {M} }}}\,\mathrm {d} \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5be0b7362b263f7a204a3700699d699c68886a65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-top: -0.34ex; width:30.825ex; height:6.176ex;" alt="{\displaystyle \quad {\overrightarrow {L_{\mathrm {O} }}}=\int _{(S)}{\overrightarrow {\mathrm {OM} }}\wedge \rho (\mathrm {M} )\,{\vec {v_{\mathrm {M} }}}\,\mathrm {d} \tau }"></span>. </p><p>Ces expressions générales ne sont guère utilisables directement. Le <a href="/wiki/Th%C3%A9or%C3%A8mes_de_K%C3%B6nig_(m%C3%A9canique)" title="Théorèmes de König (mécanique)">théorème de Koenig</a> relatif au moment cinétique permet d'en donner une forme plus compréhensible physiquement. </p> <div class="mw-heading mw-heading4"><h4 id="Théorème_de_König_(Koenig)_pour_le_moment_cinétique"><span id="Th.C3.A9or.C3.A8me_de_K.C3.B6nig_.28Koenig.29_pour_le_moment_cin.C3.A9tique"></span>Théorème de König (Koenig) pour le moment cinétique</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moment_cin%C3%A9tique&veaction=edit&section=7" title="Modifier la section : Théorème de König (Koenig) pour le moment cinétique" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Moment_cin%C3%A9tique&action=edit&section=7" title="Modifier le code source de la section : Théorème de König (Koenig) pour le moment cinétique"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Th%C3%A9or%C3%A8mes_de_K%C3%B6nig_(m%C3%A9canique)" title="Théorèmes de König (mécanique)">Théorèmes de König (mécanique)</a>.</div></div> <p>Si <i>C</i> est le <a href="/wiki/Centre_d%27inertie" title="Centre d'inertie">centre d'inertie</a> du système, et <i>M</i> la masse totale de celui-ci, alors il est possible de montrer que pour tout système matériel : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {L_{O}}}={\vec {OC}}\wedge M{\vec {v_{C}}}+{\vec {L}}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>C</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∧</mo> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {L_{O}}}={\vec {OC}}\wedge M{\vec {v_{C}}}+{\vec {L}}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc39088b1eaa24fa62428852212f5054d90ff86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.818ex; height:4.009ex;" alt="{\displaystyle {\vec {L_{O}}}={\vec {OC}}\wedge M{\vec {v_{C}}}+{\vec {L}}^{*}}"></span>,</dd></dl> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {L}}^{*}=\sum _{i}{\overrightarrow {CM}}_{i}\wedge (m_{i}{\vec {v}}_{i}^{*})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>C</mi> <mi>M</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∧</mo> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {L}}^{*}=\sum _{i}{\overrightarrow {CM}}_{i}\wedge (m_{i}{\vec {v}}_{i}^{*})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42d0115a136e1e4fec3f912fd13484b48dad1d4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-top: -0.398ex; width:24.2ex; height:6.509ex;" alt="{\displaystyle {\vec {L}}^{*}=\sum _{i}{\overrightarrow {CM}}_{i}\wedge (m_{i}{\vec {v}}_{i}^{*})}"></span> étant le moment cinétique <i>propre</i> du système, c'est-à-dire celui évalué dans le <a href="/wiki/R%C3%A9f%C3%A9rentiel_barycentrique" title="Référentiel barycentrique">référentiel barycentrique</a> (<i>R</i><sup>*</sup>) associé à (<i>R</i>), qui est le référentiel lié à <i>C</i> dont les axes sont en translation par rapport à ceux de (<i>R</i>). En utilisant les propriétés du centre d'inertie <i>C</i>, il est possible de montrer que le moment cinétique propre ne dépend <i>pas</i> du point <i>O</i> où il est évalué, et est tel que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {L}}^{*}={\vec {L}}_{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {L}}^{*}={\vec {L}}_{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a200a0e502772174a163d83f3880944951a0a01b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.8ex; height:3.343ex;" alt="{\displaystyle {\vec {L}}^{*}={\vec {L}}_{C}}"></span>. </p><p>Physiquement, le théorème de König exprime le fait que pour un système matériel, le moment cinétique par rapport à un point est la somme de celui du centre d'inertie, affecté de la masse totale du système, et du moment cinétique propre du système. Il est donc possible de séparer le mouvement du centre d'inertie du mouvement propre du système. </p><p>La séparation des deux types de moment cinétique par le théorème est assez intuitive : ainsi, pour la Terre, dans un référentiel héliocentrique, il est facile de voir que le moment cinétique se décompose en un moment cinétique « orbital » lié à son mouvement de révolution autour du <a href="/wiki/Soleil" title="Soleil">Soleil</a>, et un moment cinétique « propre », lié à sa rotation propre autour de l'axe des pôles. </p><p><span id="Tenseur_inertie"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Cas_d'un_solide_:_tenseur_d'inertie"><span id="Cas_d.27un_solide_:_tenseur_d.27inertie"></span>Cas d'un solide : tenseur d'inertie</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moment_cin%C3%A9tique&veaction=edit&section=8" title="Modifier la section : Cas d'un solide : tenseur d'inertie" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Moment_cin%C3%A9tique&action=edit&section=8" title="Modifier le code source de la section : Cas d'un solide : tenseur d'inertie"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Tenseur_d%27inertie" class="mw-redirect" title="Tenseur d'inertie">Tenseur d'inertie</a>.</div></div> <p>Dans le cas particulier d'un solide idéal, il est possible de donner une expression générale du moment cinétique propre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {L}}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {L}}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89a14caafc3164b9504127d80b3b96da5f7c8d47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:3.009ex;" alt="{\displaystyle {\vec {L}}^{*}}"></span> en fonction du <a href="/wiki/Torseur_cin%C3%A9matique" title="Torseur cinématique">vecteur rotation</a> propre du solide (<i>S</i>) par rapport au référentiel d'étude (<i>R</i>), noté <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\omega }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e066a68ceb355e3314fb2b97f1c0c421ca6074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:2.343ex;" alt="{\displaystyle {\vec {\omega }}}"></span><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite_crochet">[</span>7<span class="cite_crochet">]</span></a></sup>. </p><p>En effet, dans le cas d'un <a href="/wiki/Mod%C3%A8le_du_solide_(m%C3%A9canique)" title="Modèle du solide (mécanique)">solide</a> idéal (<i>S</i>), la vitesse de tout point <i>M<sub>i</sub></i> est donnée dans le référentiel barycentrique (<i>R</i><sup>*</sup>) par le <i>champ des vitesses</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}_{i}^{*}={\vec {\omega }}\wedge {\overrightarrow {CM_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>C</mi> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}_{i}^{*}={\vec {\omega }}\wedge {\overrightarrow {CM_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0f5fe9229d9b0403e0a284909fea8236bca77ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-top: -0.398ex; width:14.307ex; height:4.343ex;" alt="{\displaystyle {\vec {v}}_{i}^{*}={\vec {\omega }}\wedge {\overrightarrow {CM_{i}}}}"></span><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite_crochet">[</span>8<span class="cite_crochet">]</span></a></sup>. </p><p>Par suite, le moment cinétique propre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {L}}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {L}}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89a14caafc3164b9504127d80b3b96da5f7c8d47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:3.009ex;" alt="{\displaystyle {\vec {L}}^{*}}"></span> du solide (<i>S</i>) est donné par : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {L}}^{*}=\sum _{i}m_{i}{\overrightarrow {CM_{i}}}\wedge \left({\vec {\omega }}\wedge {\overrightarrow {CM_{i}}}\right)=\sum _{i}m_{i}\left(CM_{i}^{2}{\vec {\omega }}-{\overrightarrow {CM_{i}}}({\overrightarrow {CM_{i}}}\cdot {\vec {\omega }})\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>C</mi> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>→</mo> </mover> </mrow> <mo>∧</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>C</mi> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>→</mo> </mover> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>C</mi> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>C</mi> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>→</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>C</mi> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>→</mo> </mover> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {L}}^{*}=\sum _{i}m_{i}{\overrightarrow {CM_{i}}}\wedge \left({\vec {\omega }}\wedge {\overrightarrow {CM_{i}}}\right)=\sum _{i}m_{i}\left(CM_{i}^{2}{\vec {\omega }}-{\overrightarrow {CM_{i}}}({\overrightarrow {CM_{i}}}\cdot {\vec {\omega }})\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bb578bd7d77e7d82c0b8a2623543483096e2181" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:70.179ex; height:6.676ex;" alt="{\displaystyle {\vec {L}}^{*}=\sum _{i}m_{i}{\overrightarrow {CM_{i}}}\wedge \left({\vec {\omega }}\wedge {\overrightarrow {CM_{i}}}\right)=\sum _{i}m_{i}\left(CM_{i}^{2}{\vec {\omega }}-{\overrightarrow {CM_{i}}}({\overrightarrow {CM_{i}}}\cdot {\vec {\omega }})\right)}"></span>,</dd></dl> <p>en posant en coordonnées cartésiennes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\omega }}=(\omega _{x},\omega _{y},\omega _{z})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\omega }}=(\omega _{x},\omega _{y},\omega _{z})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47c8d7c011be07b5a15aed2e1edd1a6fcb5a1dc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.982ex; height:3.009ex;" alt="{\displaystyle {\vec {\omega }}=(\omega _{x},\omega _{y},\omega _{z})}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {CM_{i}}}=(x_{i},y_{i},z_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>C</mi> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>→</mo> </mover> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {CM_{i}}}=(x_{i},y_{i},z_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8cb005beccc72029fbcfea71f63797f7b1807a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-top: -0.398ex; width:17.875ex; height:4.343ex;" alt="{\displaystyle {\overrightarrow {CM_{i}}}=(x_{i},y_{i},z_{i})}"></span>, il vient pour les différentes composantes de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {L}}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {L}}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89a14caafc3164b9504127d80b3b96da5f7c8d47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:3.009ex;" alt="{\displaystyle {\vec {L}}^{*}}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}L_{x}^{*}=\omega _{x}\left(\sum _{i}m_{i}(y_{i}^{2}+z_{i}^{2})\right)-\omega _{y}\left(\sum _{i}m_{i}x_{i}y_{i}\right)-\omega _{z}\left(\sum _{i}m_{i}x_{i}z_{i}\right)\\L_{y}^{*}=-\omega _{x}\left(\sum _{i}m_{i}x_{i}y_{i}\right)+\omega _{y}\left(\sum _{i}m_{i}(x_{i}^{2}+z_{i}^{2})\right)-\omega _{z}\left(\sum _{i}m_{i}y_{i}z_{i}\right)\\L_{z}^{*}=-\omega _{x}\left(\sum _{i}m_{i}x_{i}z_{i}\right)-\omega _{y}\left(\sum _{i}m_{i}y_{i}z_{i}\right)+\omega _{z}\left(\sum _{i}m_{i}(x_{i}^{2}+y_{i}^{2})\right)\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>=</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}L_{x}^{*}=\omega _{x}\left(\sum _{i}m_{i}(y_{i}^{2}+z_{i}^{2})\right)-\omega _{y}\left(\sum _{i}m_{i}x_{i}y_{i}\right)-\omega _{z}\left(\sum _{i}m_{i}x_{i}z_{i}\right)\\L_{y}^{*}=-\omega _{x}\left(\sum _{i}m_{i}x_{i}y_{i}\right)+\omega _{y}\left(\sum _{i}m_{i}(x_{i}^{2}+z_{i}^{2})\right)-\omega _{z}\left(\sum _{i}m_{i}y_{i}z_{i}\right)\\L_{z}^{*}=-\omega _{x}\left(\sum _{i}m_{i}x_{i}z_{i}\right)-\omega _{y}\left(\sum _{i}m_{i}y_{i}z_{i}\right)+\omega _{z}\left(\sum _{i}m_{i}(x_{i}^{2}+y_{i}^{2})\right)\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e9ecc87000c1deb1afcf84d6f6a248fbb7554a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:67.033ex; height:9.843ex;" alt="{\displaystyle {\begin{cases}L_{x}^{*}=\omega _{x}\left(\sum _{i}m_{i}(y_{i}^{2}+z_{i}^{2})\right)-\omega _{y}\left(\sum _{i}m_{i}x_{i}y_{i}\right)-\omega _{z}\left(\sum _{i}m_{i}x_{i}z_{i}\right)\\L_{y}^{*}=-\omega _{x}\left(\sum _{i}m_{i}x_{i}y_{i}\right)+\omega _{y}\left(\sum _{i}m_{i}(x_{i}^{2}+z_{i}^{2})\right)-\omega _{z}\left(\sum _{i}m_{i}y_{i}z_{i}\right)\\L_{z}^{*}=-\omega _{x}\left(\sum _{i}m_{i}x_{i}z_{i}\right)-\omega _{y}\left(\sum _{i}m_{i}y_{i}z_{i}\right)+\omega _{z}\left(\sum _{i}m_{i}(x_{i}^{2}+y_{i}^{2})\right)\end{cases}}}"></span>,</dd></dl> <p>qui peut aussi s'écrire sous la forme intrinsèque : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {L}}^{*}={\bar {\bar {I}}}{\vec {\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>I</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {L}}^{*}={\bar {\bar {I}}}{\vec {\omega }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1d771eb7be5a210ca606fb421d48d0b0f448c67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.703ex; height:3.009ex;" alt="{\displaystyle {\vec {L}}^{*}={\bar {\bar {I}}}{\vec {\omega }}}"></span>,</dd></dl> <p>avec <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\bar {I}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>I</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\bar {I}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26ff8e6c0909e27d026881a560872cc0bfd6e63e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.522ex; height:2.843ex;" alt="{\displaystyle {\bar {\bar {I}}}}"></span> <i><a href="/wiki/Moment_d%27inertie" title="Moment d'inertie">tenseur d'inertie</a></i> du solide (<i>S</i>), donné par : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\bar {I}}}={\begin{bmatrix}\sum _{i}m_{i}(y_{i}^{2}+z_{i}^{2})&-\sum _{i}m_{i}x_{i}y_{i}&-\sum _{i}m_{i}x_{i}z_{i}\\-\sum _{i}m_{i}x_{i}y_{i}&\sum _{i}m_{i}(x_{i}^{2}+z_{i}^{2})&-\sum _{i}m_{i}y_{i}z_{i}\\-\sum _{i}m_{i}x_{i}z_{i}&-\sum _{i}m_{i}y_{i}z_{i}&\sum _{i}m_{i}(x_{i}^{2}+y_{i}^{2})\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>I</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\bar {I}}}={\begin{bmatrix}\sum _{i}m_{i}(y_{i}^{2}+z_{i}^{2})&-\sum _{i}m_{i}x_{i}y_{i}&-\sum _{i}m_{i}x_{i}z_{i}\\-\sum _{i}m_{i}x_{i}y_{i}&\sum _{i}m_{i}(x_{i}^{2}+z_{i}^{2})&-\sum _{i}m_{i}y_{i}z_{i}\\-\sum _{i}m_{i}x_{i}z_{i}&-\sum _{i}m_{i}y_{i}z_{i}&\sum _{i}m_{i}(x_{i}^{2}+y_{i}^{2})\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c56c77240c8cd2212fd02a8d74fef837c5bd7081" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:59.995ex; height:10.176ex;" alt="{\displaystyle {\bar {\bar {I}}}={\begin{bmatrix}\sum _{i}m_{i}(y_{i}^{2}+z_{i}^{2})&-\sum _{i}m_{i}x_{i}y_{i}&-\sum _{i}m_{i}x_{i}z_{i}\\-\sum _{i}m_{i}x_{i}y_{i}&\sum _{i}m_{i}(x_{i}^{2}+z_{i}^{2})&-\sum _{i}m_{i}y_{i}z_{i}\\-\sum _{i}m_{i}x_{i}z_{i}&-\sum _{i}m_{i}y_{i}z_{i}&\sum _{i}m_{i}(x_{i}^{2}+y_{i}^{2})\end{bmatrix}}}"></span>.</dd></dl> <p>Il en résulte qu'en général le moment cinétique propre du solide <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {L}}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {L}}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89a14caafc3164b9504127d80b3b96da5f7c8d47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:3.009ex;" alt="{\displaystyle {\vec {L}}^{*}}"></span> n'est <i>pas</i> <a href="/wiki/Colin%C3%A9arit%C3%A9" title="Colinéarité">colinéaire</a> à son vecteur rotation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\omega }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e066a68ceb355e3314fb2b97f1c0c421ca6074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:2.343ex;" alt="{\displaystyle {\vec {\omega }}}"></span> dans (<i>R</i>). </p><p>Le <i>tenseur d'inertie</i> est une caractéristique propre du solide (<i>S</i>), et donne la répartition des masses en son sein. Il s'agit d'un <a href="/wiki/Tenseur_sym%C3%A9trique" title="Tenseur symétrique">tenseur symétrique</a>. Ses éléments diagonaux sont constitués des moments d'inertie du solide par rapport aux axes (<i>Ox</i>), (<i>Oy</i>) et (<i>Oz</i>) respectivement, et ses éléments non diagonaux sont égaux à l'opposé des moments d'inertie par rapport aux plans (<i>xOy</i>), (<i>xOz</i>) et (<i>yOz</i>). </p><p>Du fait de son caractère symétrique, il est toujours possible de diagonaliser ce tenseur par un choix judicieux des axes, qui sont appelés alors <i>axes principaux d'inertie</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Moment_cinétique_et_isotropie"><span id="Moment_cin.C3.A9tique_et_isotropie"></span>Moment cinétique et isotropie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moment_cin%C3%A9tique&veaction=edit&section=9" title="Modifier la section : Moment cinétique et isotropie" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Moment_cin%C3%A9tique&action=edit&section=9" title="Modifier le code source de la section : Moment cinétique et isotropie"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La notion d'<a href="/wiki/Isotropie" title="Isotropie">isotropie</a> de l'espace traduit l'équivalence de toutes les directions dans celui-ci: dire que l'espace est isotrope signifie donc aussi qu'il est <a href="/wiki/Invariant" title="Invariant">invariant</a> par toute rotation spatiale autour d'un point quelconque. Cette propriété est en particulier valable pour un système dit <i>isolé</i>, c'est-à-dire qui n'est soumis à aucune action extérieure. </p><p>Dans le <a href="/wiki/M%C3%A9canique_hamiltonienne" title="Mécanique hamiltonienne">formalisme hamiltonien</a><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite_crochet">[</span>9<span class="cite_crochet">]</span></a></sup>, cela implique que la fonction de Hamilton <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\left({\vec {r}}_{1},\ldots ,{\vec {r}}_{n};{\vec {p}}_{1},\ldots ,{\vec {p}}_{n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>;</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\left({\vec {r}}_{1},\ldots ,{\vec {r}}_{n};{\vec {p}}_{1},\ldots ,{\vec {p}}_{n}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81b4c291c69147ff9159a8f68251c0a8fb954201" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.292ex; height:2.843ex;" alt="{\displaystyle H\left({\vec {r}}_{1},\ldots ,{\vec {r}}_{n};{\vec {p}}_{1},\ldots ,{\vec {p}}_{n}\right)}"></span> de tout système isolé de <i>n</i> points matériels soit invariante par toute rotation globale du système autour de l'origine <i>O</i> (arbitraire). En particulier, cela sera valable pour une <i>rotation élémentaire</i> arbitraire de vecteur <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {\delta \phi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>δ</mi> <mi>ϕ</mi> </mrow> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {\delta \phi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5dea495275ed742ec5dbe354497c3a823b871f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.37ex; width:2.564ex; height:4.176ex;" alt="{\displaystyle {\overrightarrow {\delta \phi }}}"></span>, telle que la variation du vecteur position <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de93eb4c8bca39012a94e9809c45d7fd677bf975" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.023ex; height:2.676ex;" alt="{\displaystyle {\vec {r}}_{i}}"></span> soit donnée par <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta {\vec {r}}_{i}={\overrightarrow {\delta \phi }}\wedge {\vec {r}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>δ</mi> <mi>ϕ</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>∧</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta {\vec {r}}_{i}={\overrightarrow {\delta \phi }}\wedge {\vec {r}}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd38bdf69eec07960a76b77d2bc95d35415e37e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.37ex; width:13.339ex; height:4.176ex;" alt="{\displaystyle \delta {\vec {r}}_{i}={\overrightarrow {\delta \phi }}\wedge {\vec {r}}_{i}}"></span>. </p><p>En <a href="/wiki/Coordonn%C3%A9es_cart%C3%A9siennes" title="Coordonnées cartésiennes">coordonnées cartésiennes</a>, et en l'absence de <a href="/wiki/Champ_%C3%A9lectromagn%C3%A9tique" title="Champ électromagnétique">champ électromagnétique</a> (ou pour une particule non chargée) les impulsions généralisées <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p}}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0f7ed082c99f239319703553a9784a0d8809768" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:2.214ex; height:2.843ex;" alt="{\displaystyle {\vec {p}}_{i}}"></span> coïncident avec les quantités de mouvement des différents points matériels <i>M<sub>i</sub></i><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite_crochet">[</span>10<span class="cite_crochet">]</span></a></sup>. Par suite <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta {\vec {p}}_{i}={\overrightarrow {\delta \phi }}\wedge {\vec {p}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>δ</mi> <mi>ϕ</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>∧</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta {\vec {p}}_{i}={\overrightarrow {\delta \phi }}\wedge {\vec {p}}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/193ea3cf0631a6cecf2a689ab4929b4ceb0457e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-top: -0.37ex; width:13.544ex; height:4.343ex;" alt="{\displaystyle \delta {\vec {p}}_{i}={\overrightarrow {\delta \phi }}\wedge {\vec {p}}_{i}}"></span>, et les équations canoniques de Hamilton se mettent sous la forme vectorielle : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}{\dot {\vec {r}}}_{i}={\vec {\nabla }}_{{\vec {p}}_{i}}H\\{\dot {\vec {p}}}_{i}=-{\vec {\nabla }}_{{\vec {r}}_{i}}H\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mi>H</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mi>H</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}{\dot {\vec {r}}}_{i}={\vec {\nabla }}_{{\vec {p}}_{i}}H\\{\dot {\vec {p}}}_{i}=-{\vec {\nabla }}_{{\vec {r}}_{i}}H\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42179e1747f8d671d1065369a3f0619d372f5314" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:15.45ex; height:7.843ex;" alt="{\displaystyle {\begin{cases}{\dot {\vec {r}}}_{i}={\vec {\nabla }}_{{\vec {p}}_{i}}H\\{\dot {\vec {p}}}_{i}=-{\vec {\nabla }}_{{\vec {r}}_{i}}H\end{cases}}}"></span>.</dd></dl> <p>La variation correspondante du hamiltonien <i>H</i> résultant de la rotation élémentaire de vecteur <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {\delta \phi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>δ</mi> <mi>ϕ</mi> </mrow> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {\delta \phi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5dea495275ed742ec5dbe354497c3a823b871f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.37ex; width:2.564ex; height:4.176ex;" alt="{\displaystyle {\overrightarrow {\delta \phi }}}"></span> peut s'exprimer par : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta H=\sum _{i=1}^{n}\left({\vec {\nabla }}_{{\vec {r}}_{i}}H\cdot (\delta {\vec {r}}_{i})+{\vec {\nabla }}_{{\vec {p}}_{i}}H\cdot (\delta {\vec {p}}_{i})\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>H</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mi>H</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mi>H</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta H=\sum _{i=1}^{n}\left({\vec {\nabla }}_{{\vec {r}}_{i}}H\cdot (\delta {\vec {r}}_{i})+{\vec {\nabla }}_{{\vec {p}}_{i}}H\cdot (\delta {\vec {p}}_{i})\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdc08db143c2992ab9c941150ca3877cb2d70465" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.4ex; height:6.843ex;" alt="{\displaystyle \delta H=\sum _{i=1}^{n}\left({\vec {\nabla }}_{{\vec {r}}_{i}}H\cdot (\delta {\vec {r}}_{i})+{\vec {\nabla }}_{{\vec {p}}_{i}}H\cdot (\delta {\vec {p}}_{i})\right)}"></span>,</dd></dl> <p>soit en tenant compte des équations de Hamilton : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta H=\sum _{i=1}^{n}\left(-{\dot {\vec {p}}}_{i}\cdot ({\overrightarrow {\delta \phi }}\wedge {\vec {r}}_{i})+{\dot {\vec {r}}}_{i}\cdot ({\overrightarrow {\delta \phi }}\wedge {\vec {p}}_{i})\right)=-{\overrightarrow {\delta \phi }}\cdot \left[\sum _{i=1}^{n}\left({\vec {r}}_{i}\wedge {\dot {\vec {p}}}_{i}+{\dot {\vec {r}}}_{i}\wedge {\vec {p}}_{i}\right)\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>H</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>δ</mi> <mi>ϕ</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>∧</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>δ</mi> <mi>ϕ</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>∧</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>δ</mi> <mi>ϕ</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>⋅</mo> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∧</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∧</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta H=\sum _{i=1}^{n}\left(-{\dot {\vec {p}}}_{i}\cdot ({\overrightarrow {\delta \phi }}\wedge {\vec {r}}_{i})+{\dot {\vec {r}}}_{i}\cdot ({\overrightarrow {\delta \phi }}\wedge {\vec {p}}_{i})\right)=-{\overrightarrow {\delta \phi }}\cdot \left[\sum _{i=1}^{n}\left({\vec {r}}_{i}\wedge {\dot {\vec {p}}}_{i}+{\dot {\vec {r}}}_{i}\wedge {\vec {p}}_{i}\right)\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90165aa9d31e18b3ab520b962489f489b1a1a3e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:78.404ex; height:7.509ex;" alt="{\displaystyle \delta H=\sum _{i=1}^{n}\left(-{\dot {\vec {p}}}_{i}\cdot ({\overrightarrow {\delta \phi }}\wedge {\vec {r}}_{i})+{\dot {\vec {r}}}_{i}\cdot ({\overrightarrow {\delta \phi }}\wedge {\vec {p}}_{i})\right)=-{\overrightarrow {\delta \phi }}\cdot \left[\sum _{i=1}^{n}\left({\vec {r}}_{i}\wedge {\dot {\vec {p}}}_{i}+{\dot {\vec {r}}}_{i}\wedge {\vec {p}}_{i}\right)\right]}"></span>,</dd></dl> <p>qui se met aussitôt sous la forme : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta H=-{\overrightarrow {\delta \phi }}\cdot \left[{\frac {\mathrm {d} }{\mathrm {d} t}}\left(\sum _{i=1}^{n}{\vec {r}}_{i}\wedge {\vec {p}}_{i}\right)\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>H</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>δ</mi> <mi>ϕ</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>⋅</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∧</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta H=-{\overrightarrow {\delta \phi }}\cdot \left[{\frac {\mathrm {d} }{\mathrm {d} t}}\left(\sum _{i=1}^{n}{\vec {r}}_{i}\wedge {\vec {p}}_{i}\right)\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ccbe02bdc56085ee68eacdc55c2d676b5661eb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.481ex; height:7.509ex;" alt="{\displaystyle \delta H=-{\overrightarrow {\delta \phi }}\cdot \left[{\frac {\mathrm {d} }{\mathrm {d} t}}\left(\sum _{i=1}^{n}{\vec {r}}_{i}\wedge {\vec {p}}_{i}\right)\right]}"></span>.</dd></dl> <p>Puisque ce résultat est valable pour toute rotation élémentaire arbitraire, l'isotropie de l'espace pour un système isolé implique alors que la quantité <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {L}}_{O}\equiv \sum _{i=1}^{n}{\vec {r}}_{i}\wedge {\vec {p}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> <mo>≡</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∧</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {L}}_{O}\equiv \sum _{i=1}^{n}{\vec {r}}_{i}\wedge {\vec {p}}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55623248beca39a97e92da62a2d3491c838180fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.64ex; height:6.843ex;" alt="{\displaystyle {\vec {L}}_{O}\equiv \sum _{i=1}^{n}{\vec {r}}_{i}\wedge {\vec {p}}_{i}}"></span>, appelé <i>moment cinétique</i> du système par rapport à l'origine <i>O</i>, soit une <i>constante du mouvement</i>. Il est clair que pour un seul point matériel, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {L}}_{O}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {L}}_{O}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c242214d1d383794d6208a4109df579ad927" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.069ex; height:3.176ex;" alt="{\displaystyle {\vec {L}}_{O}}"></span> est bien identique à la définition donnée plus haut. </p><p>Par ailleurs en procédant toujours en coordonnées cartésiennes, et compte tenu du fait que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\vec {p}}}_{i}=-{\vec {\nabla }}_{{\vec {r}}_{i}}H={\vec {F}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mi>H</mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\vec {p}}}_{i}=-{\vec {\nabla }}_{{\vec {r}}_{i}}H={\vec {F}}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5e8194c1b943df90c55913e2a6007e74a4ad983" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; margin-left: -0.089ex; width:18.583ex; height:3.843ex;" alt="{\displaystyle {\dot {\vec {p}}}_{i}=-{\vec {\nabla }}_{{\vec {r}}_{i}}H={\vec {F}}_{i}}"></span>, résultante des forces appliquées au point matériel <i>M<sub>i</sub></i>, il est possible de déduire le théorème du moment cinétique : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\vec {L}}_{O}=\sum _{i=1}^{n}\left({\vec {r}}_{i}\wedge {\dot {\vec {p}}}_{i}+{\dot {\vec {r}}}_{i}\wedge {\vec {p}}_{i}\right)=\sum _{i=1}^{n}{\vec {r}}_{i}\wedge {\vec {F}}_{i}=\sum _{i=1}^{n}{\vec {M}}_{O}({\vec {F}}_{i})={\vec {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∧</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∧</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∧</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\vec {L}}_{O}=\sum _{i=1}^{n}\left({\vec {r}}_{i}\wedge {\dot {\vec {p}}}_{i}+{\dot {\vec {r}}}_{i}\wedge {\vec {p}}_{i}\right)=\sum _{i=1}^{n}{\vec {r}}_{i}\wedge {\vec {F}}_{i}=\sum _{i=1}^{n}{\vec {M}}_{O}({\vec {F}}_{i})={\vec {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc81dad197c39760947d01ab3e0cd3aa3482543d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:66.053ex; height:6.843ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\vec {L}}_{O}=\sum _{i=1}^{n}\left({\vec {r}}_{i}\wedge {\dot {\vec {p}}}_{i}+{\dot {\vec {r}}}_{i}\wedge {\vec {p}}_{i}\right)=\sum _{i=1}^{n}{\vec {r}}_{i}\wedge {\vec {F}}_{i}=\sum _{i=1}^{n}{\vec {M}}_{O}({\vec {F}}_{i})={\vec {C}}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/757b18b7508788dc438a475d6b5868479de53d38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:3.009ex;" alt="{\displaystyle {\vec {C}}}"></span> étant la résultante du moment des forces en <i>O</i>.</dd></dl> <p>La conservation du moment cinétique par rapport à un point <i>O</i> est donc directement liée à l'invariance par rotation du hamiltonien (ou du Lagrangien) du système: c'est en particulier le cas pour un système non isolé mais soumis à un champ extérieur possédant une invariance par rotation autour de <i>O</i>. Ce type de champ très important en physique est un <i>champ à force centrale</i>, de centre de force <i>O</i> : son mouvement est alors caractérisé par la conservation du moment cinétique du système par rapport à <i>O</i>. </p><p>De même, l'invariance par rotation autour d'un axe donné du hamiltonien du système (symétrie axiale) impliquera la conservation de la composante du moment cinétique du système par rapport à cet axe, puisque alors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta H=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>H</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta H=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1898be1a61b8c29f156355a5531717e9730a6ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.373ex; height:2.343ex;" alt="{\displaystyle \delta H=0}"></span> pour toute rotation élémentaire <i>autour de cet axe</i>. Le moment résultant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/757b18b7508788dc438a475d6b5868479de53d38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:3.009ex;" alt="{\displaystyle {\vec {C}}}"></span> des forces appliquée en <i>O</i>, donc la dérivée du moment cinétique en ce même point, est, elle, directement liée à la variation du hamiltonien dans une rotation élémentaire du système d'un angle <i>δϕ</i> autour de <i>O</i>. </p><p>Enfin, il est possible en utilisant les <a href="/wiki/Crochet_de_Poisson" title="Crochet de Poisson">crochets de Poisson</a> de montrer la relation suivante entre les composantes cartésiennes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{Oi}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{Oi}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d67d70a9bae01d85cc755ffe4ca1590a19793c5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.636ex; height:2.509ex;" alt="{\displaystyle L_{Oi}}"></span> du moment cinétique, en posant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{O1}=L_{x},\,L_{O2}=L_{y},\,L_{O3}=L_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{O1}=L_{x},\,L_{O2}=L_{y},\,L_{O3}=L_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70be673ae183990ac3086c032348e3f36efe13c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.782ex; height:2.843ex;" alt="{\displaystyle L_{O1}=L_{x},\,L_{O2}=L_{y},\,L_{O3}=L_{z}}"></span> : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{L_{Oi},L_{Oj}\}=\varepsilon _{ijk}L_{Ok}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> <mi>j</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{L_{Oi},L_{Oj}\}=\varepsilon _{ijk}L_{Ok}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c27f5f310f74060ca5be2298eb89bc09672fda2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.183ex; height:3.009ex;" alt="{\displaystyle \{L_{Oi},L_{Oj}\}=\varepsilon _{ijk}L_{Ok}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ijk}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ijk}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21525193117bdfc0f3ac71b8ec46e3b6d0637daf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.417ex; height:2.343ex;" alt="{\displaystyle \varepsilon _{ijk}}"></span> étant le <a href="/wiki/Symbole_de_Levi-Civita" title="Symbole de Levi-Civita">symbole de Levi-Civita</a>.</dd></dl> <p>Cette relation à une forme très proche de celle de la relation de commutation des opérateurs de <a href="/wiki/Moment_cin%C3%A9tique_quantique" class="mw-redirect" title="Moment cinétique quantique">moment cinétique en mécanique quantique</a> : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\hat {L}}_{i},{\hat {L}}_{j}\right]=i\hbar \varepsilon _{ijk}{\hat {L}}_{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\hat {L}}_{i},{\hat {L}}_{j}\right]=i\hbar \varepsilon _{ijk}{\hat {L}}_{k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee07d782051c2cf4b16e72359a3e0d6b5c25fbff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.047ex; height:4.843ex;" alt="{\displaystyle \left[{\hat {L}}_{i},{\hat {L}}_{j}\right]=i\hbar \varepsilon _{ijk}{\hat {L}}_{k}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Moment_cinétique_et_mouvement_à_force_centrale"><span id="Moment_cin.C3.A9tique_et_mouvement_.C3.A0_force_centrale"></span>Moment cinétique et mouvement à <a href="/wiki/Force_centrale" title="Force centrale">force centrale</a></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moment_cin%C3%A9tique&veaction=edit&section=10" title="Modifier la section : Moment cinétique et mouvement à force centrale" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Moment_cin%C3%A9tique&action=edit&section=10" title="Modifier le code source de la section : Moment cinétique et mouvement à force centrale"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Mouvement_%C3%A0_force_centrale" title="Mouvement à force centrale">mouvement à force centrale</a>.</div></div> <div class="mw-heading mw-heading3"><h3 id="Cas_général"><span id="Cas_g.C3.A9n.C3.A9ral"></span>Cas général</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moment_cin%C3%A9tique&veaction=edit&section=11" title="Modifier la section : Cas général" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Moment_cin%C3%A9tique&action=edit&section=11" title="Modifier le code source de la section : Cas général"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Notion_de_force_centrale">Notion de force centrale</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moment_cin%C3%A9tique&veaction=edit&section=12" title="Modifier la section : Notion de force centrale" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Moment_cin%C3%A9tique&action=edit&section=12" title="Modifier le code source de la section : Notion de force centrale"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/BehoudImpulsmoment.ogv/250px--BehoudImpulsmoment.ogv.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="250" height="188" data-durationhint="18" data-mwtitle="BehoudImpulsmoment.ogv" data-mwprovider="wikimediacommons" resource="/wiki/Fichier:BehoudImpulsmoment.ogv"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/18/BehoudImpulsmoment.ogv/BehoudImpulsmoment.ogv.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="384" data-height="288" /><source src="//upload.wikimedia.org/wikipedia/commons/1/18/BehoudImpulsmoment.ogv" type="video/ogg; codecs="theora"" data-width="384" data-height="288" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/18/BehoudImpulsmoment.ogv/BehoudImpulsmoment.ogv.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="192" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/18/BehoudImpulsmoment.ogv/BehoudImpulsmoment.ogv.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="320" data-height="240" /></video></span><figcaption>Démonstration du moment angulaire</figcaption></figure> <p>Un cas particulier très important d'utilisation du moment cinétique est celui du <i>mouvement à <a href="/wiki/Force_centrale" title="Force centrale">force centrale</a></i>, pour lequel le moment cinétique est conservé. </p><p>La notion de force centrale est définie de diverses façons suivant les auteurs : </p> <ul><li>soit comme une force <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef40edff397a115ecdce7d3518001dfcc7f37d9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.771ex; height:2.843ex;" alt="{\displaystyle {\vec {F}}}"></span> dont la direction passe par un point <b>fixe</b> dans (<i>R</i>), appelé <i>centre de force</i>, donc en posant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}={\overrightarrow {OM}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>M</mi> </mrow> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}={\overrightarrow {OM}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7015d3f60deb3a3e3f0c5691e8f12d8c1b0b161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-top: -0.4ex; width:8.789ex; height:3.843ex;" alt="{\displaystyle {\vec {r}}={\overrightarrow {OM}}}"></span> telle que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}=F{\frac {\vec {r}}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}=F{\frac {\vec {r}}{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cb5b58dca016e4d172cdf6a1775e41a53a05991" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.669ex; height:5.343ex;" alt="{\displaystyle {\vec {F}}=F{\frac {\vec {r}}{r}}}"></span> à tout instant, <i>F</i> étant <i>quelconque</i> : il s'agit donc d'une définition purement géométrique ;</li> <li>soit comme une force dont non seulement la direction passe à tout instant par un point fixe <i>O</i>, mais qui dérive d'un potentiel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=V(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=V(r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6e702272136c9133b934e9efb66f6c50bce9665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.531ex; height:2.843ex;" alt="{\displaystyle V=V(r)}"></span> ne dépendant que de la distance au centre de force : il s'agit donc d'une force également <i>conservative</i>.</li></ul> <p>Si en pratique les cas de <a href="/wiki/Mouvement_%C3%A0_force_centrale" title="Mouvement à force centrale">mouvement à force centrale</a> se limitent le plus souvent à des forces conservatives (gravitation par exemple), il est utile de distinguer les deux notions de force <i>centrale</i> et de force <i>conservative</i>. Aussi une force sera considérée comme <i><b>centrale</b></i> si à tout instant <i>sa direction passe par un point fixe </i>O<i>, qu'elle soit ou non conservative</i>. </p><p>Ceci permet de distinguer dans les conséquences du caractère <i>central</i> de la force ce qui est lié à l'aspect géométrique (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef40edff397a115ecdce7d3518001dfcc7f37d9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.771ex; height:2.843ex;" alt="{\displaystyle {\vec {F}}}"></span> est toujours colinéraire à <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aec3c9ce13b53e9e24c98e7cce4212627884c91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}"></span>) de ce qui est lié à un éventuel caractère conservatif, donc le fait que l'énergie <a href="/wiki/M%C3%A9canique_du_point" title="Mécanique du point">mécanique du point</a> matériel est conservée. </p><p>Ainsi si toute force dérivant d'un potentiel scalaire dépendant uniquement de la distance <i>r</i> à l'origine est centrale, une force non conservative a priori comme la tension du fil d'un <a href="/wiki/Pendule_simple" title="Pendule simple">pendule simple</a>, qui pointe à tout instant vers le point de fixation du pendule, sera également considérée comme une force centrale avec cette définition<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite_crochet">[</span>11<span class="cite_crochet">]</span></a></sup>. </p> <div class="mw-heading mw-heading4"><h4 id="Conservation_du_moment_cinétique_et_planéité_de_la_trajectoire"><span id="Conservation_du_moment_cin.C3.A9tique_et_plan.C3.A9it.C3.A9_de_la_trajectoire"></span>Conservation du moment cinétique et planéité de la trajectoire</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moment_cin%C3%A9tique&veaction=edit&section=13" title="Modifier la section : Conservation du moment cinétique et planéité de la trajectoire" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Moment_cin%C3%A9tique&action=edit&section=13" title="Modifier le code source de la section : Conservation du moment cinétique et planéité de la trajectoire"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Conservation_du_moment_cin%C3%A9tique" title="Conservation du moment cinétique">Conservation du moment cinétique</a>.</div></div> <p>Le mouvement d'un point matériel <i>M</i> sous le seul effet d'une force centrale est un exemple de mouvement pour lequel le moment cinétique par rapport au centre de force est conservé<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite_crochet">[</span>12<span class="cite_crochet">]</span></a></sup>. En effet, en prenant pour origine le centre de force <i>O</i>, le théorème du moment cinétique donne : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\vec {\mathrm {d} L_{O}}}{\mathrm {d} t}}={\overrightarrow {OM}}\wedge {\vec {F}}={\vec {0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>M</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\vec {\mathrm {d} L_{O}}}{\mathrm {d} t}}={\overrightarrow {OM}}\wedge {\vec {F}}={\vec {0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f65adf99b8bef23cfee1fd8def1584268096bbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.378ex; height:7.009ex;" alt="{\displaystyle {\frac {\vec {\mathrm {d} L_{O}}}{\mathrm {d} t}}={\overrightarrow {OM}}\wedge {\vec {F}}={\vec {0}}}"></span>, puisque <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {OM}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>M</mi> </mrow> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {OM}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8a35bb30cf6e3998acb8ceeef6b2d23696a359e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-top: -0.4ex; width:4.468ex; height:3.843ex;" alt="{\displaystyle {\overrightarrow {OM}}}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef40edff397a115ecdce7d3518001dfcc7f37d9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.771ex; height:2.843ex;" alt="{\displaystyle {\vec {F}}}"></span> sont à tout instant colinéaires si la force est centrale.</dd></dl> <p>Par suite le moment cinétique <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {L_{O}}}={\vec {r}}\wedge {\vec {p}}={\vec {cte}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>c</mi> <mi>t</mi> <mi>e</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {L_{O}}}={\vec {r}}\wedge {\vec {p}}={\vec {cte}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10534c607f641ce0d29987f82ac335932c4be5e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.327ex; height:4.009ex;" alt="{\displaystyle {\vec {L_{O}}}={\vec {r}}\wedge {\vec {p}}={\vec {cte}}}"></span> est une <a href="/wiki/Loi_de_conservation" title="Loi de conservation">intégrale première</a> du mouvement, et donc le vecteur position <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aec3c9ce13b53e9e24c98e7cce4212627884c91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}"></span> et la <a href="/wiki/Quantit%C3%A9_de_mouvement" title="Quantité de mouvement">quantité de mouvement</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84fee53c81592db54e0fe6c6f9eba002bb1dc74b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}"></span> du corps sont à tout instant perpendiculaires à un vecteur de direction constante : <i><b>la trajectoire est donc plane</b></i>, entièrement contenue dans le plan perpendiculaire à <font style="vertical-align:+25%;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {L_{O}}}={\vec {r_{0}}}\wedge {\vec {p_{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {L_{O}}}={\vec {r_{0}}}\wedge {\vec {p_{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d714758415d4a9e7633c8e16960d860033f40ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.398ex; height:4.009ex;" alt="{\displaystyle {\vec {L_{O}}}={\vec {r_{0}}}\wedge {\vec {p_{0}}}}"></span></font> (l'indice « 0 » désigne les valeurs initiales des grandeurs). </p><p>Le mouvement ne comportant que deux <a href="/wiki/Degr%C3%A9_de_libert%C3%A9_(physique)" title="Degré de liberté (physique)">degrés de liberté</a>, il est possible de se placer en <a href="/wiki/Coordonn%C3%A9es_polaires" title="Coordonnées polaires">coordonnées polaires</a> (<i>r</i>,<i>θ</i>) dans le plan de la <a href="/wiki/Trajectoire" title="Trajectoire">trajectoire</a>. Il vient ainsi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {L_{O}}}=L{\vec {e_{z}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {L_{O}}}=L{\vec {e_{z}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ae15077d169f7aef748c447b039193df90c62b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.074ex; height:4.009ex;" alt="{\displaystyle {\vec {L_{O}}}=L{\vec {e_{z}}}}"></span>, avec <font style="vertical-align:+25%;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L\equiv mr^{2}{\dot {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>≡</mo> <mi>m</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L\equiv mr^{2}{\dot {\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8394d7f3dc635272b9902528ea7e5b72f64f2425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.181ex; height:2.843ex;" alt="{\displaystyle L\equiv mr^{2}{\dot {\theta }}}"></span></font> <b>constante</b>. </p> <div class="mw-heading mw-heading4"><h4 id="Loi_des_aires_et_formule_de_Binet">Loi des aires et formule de Binet</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moment_cin%C3%A9tique&veaction=edit&section=14" title="Modifier la section : Loi des aires et formule de Binet" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Moment_cin%C3%A9tique&action=edit&section=14" title="Modifier le code source de la section : Loi des aires et formule de Binet"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La conservation du moment cinétique par rapport au centre de force <i>O</i> peut s'interpréter physiquement par le fait que non seulement la trajectoire est plane mais également que le <a href="/wiki/Vecteur_position" title="Vecteur position">vecteur position</a> du point matériel <i>balaie des aires égales en des temps égaux</i>, autrement dit que le mouvement vérifie la <i><b>loi des aires</b></i>, mis en évidence par <a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Kepler</a> en 1609 dans le cas du mouvement des planètes (cf. <a href="/wiki/Lois_de_Kepler" title="Lois de Kepler">loi de Kepler</a>)<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite_crochet">[</span>13<span class="cite_crochet">]</span></a></sup>. </p><p>En effet en posant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=r^{2}{\dot {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=r^{2}{\dot {\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4308bebcd725680d0826a5addc940fba2e36c877" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.324ex; height:2.843ex;" alt="{\displaystyle C=r^{2}{\dot {\theta }}}"></span> (= constante), l'aire élémentaire <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d{\mathcal {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d{\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07f8a04ad70c290abef2228947a3d231b9a03baa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.119ex; height:2.343ex;" alt="{\displaystyle d{\mathcal {A}}}"></span> balayée par <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aec3c9ce13b53e9e24c98e7cce4212627884c91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}"></span> pendant la durée <i>dt</i> s'écrit <font style="vertical-align:+15%;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d{\mathcal {A}}={\tfrac {1}{2}}r^{2}d\theta ={\tfrac {1}{2}}Cdt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>C</mi> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d{\mathcal {A}}={\tfrac {1}{2}}r^{2}d\theta ={\tfrac {1}{2}}Cdt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3d783f07928f6179c68977bca6d2e543e6e8e02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.863ex; height:3.509ex;" alt="{\displaystyle d{\mathcal {A}}={\tfrac {1}{2}}r^{2}d\theta ={\tfrac {1}{2}}Cdt}"></span></font>. Par suite le taux de variation de cette aire balayée, appelé <a href="/wiki/Vitesse_ar%C3%A9olaire" title="Vitesse aréolaire">vitesse aréolaire</a> <font style="vertical-align:+15%;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\mathrm {d} {\mathcal {A}}}{\mathrm {d} t}}=C/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mstyle> </mrow> <mo>=</mo> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\mathrm {d} {\mathcal {A}}}{\mathrm {d} t}}=C/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13f2818194cce0c720f9ce467703efd274e2b5fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.286ex; height:3.843ex;" alt="{\displaystyle {\tfrac {\mathrm {d} {\mathcal {A}}}{\mathrm {d} t}}=C/2}"></span></font> est bien une constante, la quantité <i>C</i> est souvent appelée pour cette raison <i><a href="/wiki/Constante_des_aires" title="Constante des aires">constante des aires</a></i>. </p><p>Par ailleurs, le fait que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=r^{2}{\dot {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=r^{2}{\dot {\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4308bebcd725680d0826a5addc940fba2e36c877" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.324ex; height:2.843ex;" alt="{\displaystyle C=r^{2}{\dot {\theta }}}"></span> soit une constante du mouvement permet d'exprimer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f78124b2aea53527d3e053cbbdd9c7ded2c8f05f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.356ex; height:2.843ex;" alt="{\displaystyle {\dot {\theta }}}"></span> sous la forme <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\theta }}=u^{2}C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\theta }}=u^{2}C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3595f2a9d482591c105fcd6f175d3a8668e48326" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.605ex; height:2.843ex;" alt="{\displaystyle {\dot {\theta }}=u^{2}C}"></span> avec <i>u=1/r</i>. On peut alors éliminer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f78124b2aea53527d3e053cbbdd9c7ded2c8f05f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.356ex; height:2.843ex;" alt="{\displaystyle {\dot {\theta }}}"></span> dans les formules <a href="/wiki/Cin%C3%A9matique_du_point" title="Cinématique du point">cinématiques</a> donnant la vitesse et l'accélération du point matériel en coordonnées polaires, ce qui conduit à établir les <a href="/wiki/Formules_de_Binet" title="Formules de Binet">deux formules de Binet</a>. En particulier il est possible de démontrer que l'accélération du point matériel se met alors sous la forme : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}_{M}=-u^{2}C^{2}\left[{\frac {\mathrm {d} ^{2}u}{\mathrm {d} \theta ^{2}}}+u\right]{\vec {e}}_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>u</mi> </mrow> <mo>]</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}_{M}=-u^{2}C^{2}\left[{\frac {\mathrm {d} ^{2}u}{\mathrm {d} \theta ^{2}}}+u\right]{\vec {e}}_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e2f9cee7149b3624d61daf79fb0b41abf4bfbce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.44ex; height:6.343ex;" alt="{\displaystyle {\vec {a}}_{M}=-u^{2}C^{2}\left[{\frac {\mathrm {d} ^{2}u}{\mathrm {d} \theta ^{2}}}+u\right]{\vec {e}}_{r}}"></span>, (<abbr class="abbr" title="Deuxième">2<sup>e</sup></abbr> de formule de Binet).</dd></dl> <p>Bien entendu, cette formule montre bien que l'accélération est dirigée vers le centre de force, puisque la force l'est, comme le prévoit la <a href="/wiki/Lois_du_mouvement_de_Newton" title="Lois du mouvement de Newton">relation fondamentale de la dynamique</a>. Il est utile de souligner que la loi des aires comme les formules de Binet sont des conséquences du seul caractère <i>central</i> de la force, et n'implique pas que celle-ci soit conservative. </p> <div class="mw-heading mw-heading4"><h4 id="Séparation_radiale-angulaire_de_l'énergie_cinétique_et_barrière_centrifuge"><span id="S.C3.A9paration_radiale-angulaire_de_l.27.C3.A9nergie_cin.C3.A9tique_et_barri.C3.A8re_centrifuge"></span>Séparation radiale-angulaire de l'énergie cinétique et barrière centrifuge</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moment_cin%C3%A9tique&veaction=edit&section=15" title="Modifier la section : Séparation radiale-angulaire de l'énergie cinétique et barrière centrifuge" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Moment_cin%C3%A9tique&action=edit&section=15" title="Modifier le code source de la section : Séparation radiale-angulaire de l'énergie cinétique et barrière centrifuge"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Compte tenu du fait que <font style="vertical-align:+25%;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v^{2}={\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v^{2}={\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66b1a08e3131c3f1a44f9329a86c7bf663bcf798" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.98ex; height:3.343ex;" alt="{\displaystyle v^{2}={\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}}"></span></font> en coordonnées polaires, l'<a href="/wiki/%C3%89nergie_cin%C3%A9tique" title="Énergie cinétique">énergie cinétique</a> du point matériel peut dans le cas d'un mouvement à force centrale se séparer en une partie dite <i>radiale</i> et une partie dite <i>angulaire</i>. En effet il vient aussitôt : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{k}={\frac {1}{2}}m{\dot {r}}^{2}+{\frac {L^{2}}{2mr^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{k}={\frac {1}{2}}m{\dot {r}}^{2}+{\frac {L^{2}}{2mr^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61bfb0671f21dc61aab2d414018044f1ec2c6805" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.27ex; height:6.009ex;" alt="{\displaystyle E_{k}={\frac {1}{2}}m{\dot {r}}^{2}+{\frac {L^{2}}{2mr^{2}}}}"></span>,</dd></dl> <p>le premier terme est identique à celui qu'aurait l'énergie cinétique du point matériel s'il se déplaçait à vitesse <font style="vertical-align:+25%;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v={\dot {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\dot {r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa839ca98363ab4d436055c7d6378e33b251331b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.518ex; height:2.176ex;" alt="{\displaystyle v={\dot {r}}}"></span></font> le long de la direction <font style="vertical-align:+25%;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dad35ca61d14fd0fe0f53eaf14023bbe79e62f1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.197ex; height:2.676ex;" alt="{\displaystyle {\vec {e}}_{r}}"></span></font>, et donc est correspond à l’énergie cinétique radiale. Le second terme, dit angulaire de par son lien avec le mouvement orthoradial, correspond plutôt à une énergie potentielle <i>répulsive</i> en 1/<i>r</i><sup>2</sup>, dans la mesure où <i>L</i> est constant, pour un point matériel considéré comme en mouvement unidimensionnel selon la direction radiale. Ce terme est souvent appelé <i>barrière centrifuge</i>. </p><p>Si là encore la séparation radiale-angulaire de l'énergie cinétique est uniquement la conséquence du caractère central de la force et ne nécessite nullement que celle-ci soit conservative, elle a une importance particulière dans ce dernier cas, car il est alors possible de se ramener à un mouvement unidimensionnel. </p> <div class="mw-heading mw-heading3"><h3 id="Cas_où_la_force_centrale_dérive_d'une_énergie_potentielle"><span id="Cas_o.C3.B9_la_force_centrale_d.C3.A9rive_d.27une_.C3.A9nergie_potentielle"></span>Cas où la force centrale dérive d'une <a href="/wiki/%C3%89nergie_potentielle" title="Énergie potentielle">énergie potentielle</a></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moment_cin%C3%A9tique&veaction=edit&section=16" title="Modifier la section : Cas où la force centrale dérive d'une énergie potentielle" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Moment_cin%C3%A9tique&action=edit&section=16" title="Modifier le code source de la section : Cas où la force centrale dérive d'une énergie potentielle"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Probl%C3%A8me_%C3%A0_deux_corps" title="Problème à deux corps">problème à deux corps</a>.</div></div> <p>Si la force centrale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef40edff397a115ecdce7d3518001dfcc7f37d9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.771ex; height:2.843ex;" alt="{\displaystyle {\vec {F}}}"></span> dérive d'une <a href="/wiki/%C3%89nergie_potentielle" title="Énergie potentielle">énergie potentielle</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/114fdc48547ee60d02d7a2f4765d52a6ee3507d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.645ex; height:2.843ex;" alt="{\displaystyle V(r)}"></span>, l'<a href="/wiki/%C3%89nergie_m%C3%A9canique" title="Énergie mécanique">énergie mécanique</a> du corps se met sous la forme: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{m}={\frac {1}{2}}m{\dot {r}}^{2}+U_{\text{eff}}(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>eff</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{m}={\frac {1}{2}}m{\dot {r}}^{2}+U_{\text{eff}}(r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1384bbcda9c9980c5fe9596b7b55f56a86167904" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.128ex; height:5.176ex;" alt="{\displaystyle E_{m}={\frac {1}{2}}m{\dot {r}}^{2}+U_{\text{eff}}(r)}"></span> avec <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\text{eff}}(r)\equiv V(r)+{\frac {L^{2}}{2mr^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>eff</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\text{eff}}(r)\equiv V(r)+{\frac {L^{2}}{2mr^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f84ae0d5bc86dae0aad2c47cba5e6daba41f404" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:23.14ex; height:6.009ex;" alt="{\displaystyle U_{\text{eff}}(r)\equiv V(r)+{\frac {L^{2}}{2mr^{2}}}}"></span>, <a href="/wiki/%C3%89nergie_potentielle" title="Énergie potentielle">énergie potentielle</a> <b>effective</b>. </p><p>Le problème se réduit alors à un mouvement unidimensionnel d'une particule <i>fictive</i> dans un <a href="/wiki/Force_conservative" title="Force conservative">potentiel</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\text{eff}}(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>eff</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\text{eff}}(r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dcda000d7d7127b6ed8bc19ae03b4bcfab6cbfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.414ex; height:2.843ex;" alt="{\displaystyle U_{\text{eff}}(r)}"></span>. Le terme <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {L^{2}}{2mr^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {L^{2}}{2mr^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d51bf8b2f96ec00c365e91720ae5518f4fe67ea9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:6.142ex; height:6.009ex;" alt="{\displaystyle {\frac {L^{2}}{2mr^{2}}}}"></span> étant positif et croissant à courte distance, il joue le rôle de « <a href="/wiki/Barri%C3%A8re_de_potentiel" title="Barrière de potentiel">barrière de potentiel</a> <a href="/wiki/Centrifuge" class="mw-redirect" title="Centrifuge">centrifuge</a> ». La nature des mouvements possibles dépend alors du potentiel <i>V</i>(<i>r</i>) ainsi que de l'énergie mécanique totale du point matériel. </p><p>En général les trajectoires obtenues pour une <a href="/wiki/%C3%89nergie_potentielle" title="Énergie potentielle">énergie potentielle</a> <i>V</i>(<i>r</i>) quelconque ne sont <b>pas</b> des courbes fermées : seuls le potentiel <b>coulombien</b> attractif <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(r)=-{\frac {K}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>K</mi> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(r)=-{\frac {K}{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ece753c0f4de25177c658ba433562e7da34c7c55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.454ex; height:5.176ex;" alt="{\displaystyle V(r)=-{\frac {K}{r}}}"></span> (<i>K</i> constante) et le potentiel <b>harmonique</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(r)=\alpha r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>α</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(r)=\alpha r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fb35b8fcd04a50499a137b4b2871cb07d36f2df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.334ex; height:3.176ex;" alt="{\displaystyle V(r)=\alpha r^{2}}"></span> en donneront (<a href="/wiki/Th%C3%A9or%C3%A8me_de_Bertrand" title="Théorème de Bertrand">Théorème de Bertrand</a>)<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite_crochet">[</span>14<span class="cite_crochet">]</span></a></sup> (cf. <a href="/wiki/Probl%C3%A8me_%C3%A0_deux_corps" title="Problème à deux corps">problème à deux corps</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Moment_cinétique_relativiste"><span id="Moment_cin.C3.A9tique_relativiste"></span>Moment cinétique relativiste</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moment_cin%C3%A9tique&veaction=edit&section=17" title="Modifier la section : Moment cinétique relativiste" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Moment_cin%C3%A9tique&action=edit&section=17" title="Modifier le code source de la section : Moment cinétique relativiste"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La notion de moment cinétique peut être définie dans le cadre de la théorie de la <a href="/wiki/Relativit%C3%A9_restreinte" title="Relativité restreinte">relativité restreinte</a>, sous la forme d'un <a href="/wiki/Tenseur" title="Tenseur">tenseur</a> antisymétrique<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite_crochet">[</span>15<span class="cite_crochet">]</span></a></sup>. </p><p>Dans le domaine relativiste, il n'est pas possible de considérer les coordonnées d'espace indépendamment du temps, et aux vecteurs position <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aec3c9ce13b53e9e24c98e7cce4212627884c91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}"></span> et <a href="/wiki/Quantit%C3%A9_de_mouvement" title="Quantité de mouvement">quantité de mouvement</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}=m{\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p}}=m{\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e41e3b934ccee493856c6caceb07f9a14013b86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.729ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}=m{\vec {v}}}"></span> de la <a href="/wiki/M%C3%A9canique_newtonienne" title="Mécanique newtonienne">mécanique newtonienne</a>, correspondent deux <a href="/wiki/Quadrivecteur" title="Quadrivecteur">quadrivecteurs</a> : </p> <ul><li>le quadrivecteur position-temps, noté <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }"></span>, avec <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} =(ct,{\vec {r}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {X} =(ct,{\vec {r}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a506baed6f92534282d1ebb5e479817cb7a31dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.031ex; height:2.843ex;" alt="{\displaystyle \mathbf {X} =(ct,{\vec {r}})}"></span> ;</li> <li>le <a href="/wiki/Quadrivecteur_impulsion-%C3%A9nergie" class="mw-redirect" title="Quadrivecteur impulsion-énergie">quadrivecteur impulsion-énergie</a>, noté <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c250ef2a112c86b93c637dfa288c6d7f34ac3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle \mathbf {P} }"></span>, avec <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} =\left({\frac {E}{c}},\mathbf {p} \right)=(\gamma mc,\gamma m{\vec {v}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mi>c</mi> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>γ</mi> <mi>m</mi> <mi>c</mi> <mo>,</mo> <mi>γ</mi> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} =\left({\frac {E}{c}},\mathbf {p} \right)=(\gamma mc,\gamma m{\vec {v}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43001b4d40fb7bd0ea833fb89148ad375fbe3dc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.207ex; height:6.176ex;" alt="{\displaystyle \mathbf {P} =\left({\frac {E}{c}},\mathbf {p} \right)=(\gamma mc,\gamma m{\vec {v}})}"></span>, où <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\gamma mc^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>γ</mi> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=\gamma mc^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8da89dffbbabc8a3da04c57de204b0f8590ab99f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.238ex; height:3.176ex;" alt="{\displaystyle E=\gamma mc^{2}}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} =\gamma m{\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mi>γ</mi> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =\gamma m{\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a8801642fc632afa3baed95ccc5c95857715e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.062ex; height:2.843ex;" alt="{\displaystyle \mathbf {p} =\gamma m{\vec {v}}}"></span> correspondent respectivement à l'énergie et la quantité de mouvement relativistes de la particule. Le quadrivecteur impulsion-énergie est en fait donné par <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} =m\mathbf {U} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} =m\mathbf {U} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a02705be15bd1f3bb7e3e40900da6237fb69c1d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.022ex; height:2.176ex;" alt="{\displaystyle \mathbf {P} =m\mathbf {U} }"></span> où <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {U} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {U} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2141bec2344e3dc5241ff50b0fd366755e00223" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.057ex; height:2.176ex;" alt="{\displaystyle \mathbf {U} }"></span> est le quadrivecteur vitesse, défini par <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {U} ={\frac {\mathrm {d} \mathbf {X} }{\mathrm {d} \tau }}=\gamma {\frac {\mathrm {d} \mathbf {X} }{\mathrm {d} t}}=(\gamma c,\gamma {\vec {v}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>γ</mi> <mi>c</mi> <mo>,</mo> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {U} ={\frac {\mathrm {d} \mathbf {X} }{\mathrm {d} \tau }}=\gamma {\frac {\mathrm {d} \mathbf {X} }{\mathrm {d} t}}=(\gamma c,\gamma {\vec {v}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d3a465bae8688965dfe9a03040536a5b94941dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.461ex; height:5.509ex;" alt="{\displaystyle \mathbf {U} ={\frac {\mathrm {d} \mathbf {X} }{\mathrm {d} \tau }}=\gamma {\frac {\mathrm {d} \mathbf {X} }{\mathrm {d} t}}=(\gamma c,\gamma {\vec {v}})}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}"></span> étant le vecteur vitesse « ordinaire » de la particule, et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> le <a href="/wiki/Temps_propre" title="Temps propre">temps propre</a>.</li></ul> <p>Les composantes de ces deux quadrivecteurs étant notées <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc55f5e6e0d894581b9a8c509fc67ab8579789e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.281ex; height:2.343ex;" alt="{\displaystyle X^{\alpha }}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7c29c141756ef30b7a902b2778fe15bbcd9c58e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.106ex; height:2.343ex;" alt="{\displaystyle P^{\alpha }}"></span> avec <span class="nowrap"><i>α</i> = 0,1,2,3</span>, il est possible de définir un (quadri)tenseur contravariant antisymétrique du second ordre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e499ae5946af9c09777ada933051b3669d3372c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.537ex; height:2.176ex;" alt="{\displaystyle \mathbf {M} }"></span> appelé (quadri)tenseur moment cinétique dont les composantes sont données par : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{\alpha \beta }=X^{\alpha }P^{\beta }-X^{\beta }P^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M^{\alpha \beta }=X^{\alpha }P^{\beta }-X^{\beta }P^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00abd5f409fd7371f8a0cc13306a25fde243ac54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:23.217ex; height:2.843ex;" alt="{\displaystyle M^{\alpha \beta }=X^{\alpha }P^{\beta }-X^{\beta }P^{\alpha }}"></span>,</dd></dl> <p>du fait de son caractère antisymétrique, ce tenseur ne possède en réalité que 6 composantes indépendantes, trois mixtes (<i>M</i><sup>01</sup>, <i>M</i><sup>02</sup> et <i>M</i><sup>03</sup>) et trois autres du genre espace (<i>M</i><sup>12</sup>, <i>M</i><sup>23</sup> et <i>M</i><sup>13</sup>). En explicitant ces dernières, il vient aussitôt : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}M^{12}=\gamma m\left(xv_{y}-v_{x}y\right)=\gamma L_{z}\\M^{13}=\gamma m\left(xv_{z}-v_{x}z\right)=-\gamma L_{y}\\M^{23}=\gamma m\left(yv_{z}-v_{y}z\right)=\gamma L_{x}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msup> <mo>=</mo> <mi>γ</mi> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>γ</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msup> <mo>=</mo> <mi>γ</mi> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mi>γ</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msup> <mo>=</mo> <mi>γ</mi> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <mi>y</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>γ</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}M^{12}=\gamma m\left(xv_{y}-v_{x}y\right)=\gamma L_{z}\\M^{13}=\gamma m\left(xv_{z}-v_{x}z\right)=-\gamma L_{y}\\M^{23}=\gamma m\left(yv_{z}-v_{y}z\right)=\gamma L_{x}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1196979e84f534578a89a86cb3863a72c9f7f86e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.821ex; margin-bottom: -0.183ex; width:34.279ex; height:9.176ex;" alt="{\displaystyle {\begin{cases}M^{12}=\gamma m\left(xv_{y}-v_{x}y\right)=\gamma L_{z}\\M^{13}=\gamma m\left(xv_{z}-v_{x}z\right)=-\gamma L_{y}\\M^{23}=\gamma m\left(yv_{z}-v_{y}z\right)=\gamma L_{x}\end{cases}}}"></span>,</dd></dl> <p><br /> où <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{x},L_{y},L_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{x},L_{y},L_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c960a840c2fffc041ef8845ef56a9dfdfc203964" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.04ex; height:2.843ex;" alt="{\displaystyle L_{x},L_{y},L_{z}}"></span> représentent les composantes cartésiennes du moment cinétique non relativiste. Dans la limite des faibles vitesses devant <span class="nowrap"><i>c</i>, <i>γ</i> → 1</span> et les trois composantes spatiales indépendantes du tenseur coïncident, au signe près, à celle du moment cinétique « ordinaire ». </p> <div class="mw-heading mw-heading2"><h2 id="Notes_et_références"><span id="Notes_et_r.C3.A9f.C3.A9rences"></span>Notes et références</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moment_cin%C3%A9tique&veaction=edit&section=18" title="Modifier la section : Notes et références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Moment_cin%C3%A9tique&action=edit&section=18" title="Modifier le code source de la section : Notes et références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small decimal" style=""><div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink noprint"><a href="#cite_ref-1">↑</a> </span><span class="reference-text">En général, le moment cinétique d'un ensemble de points n'est pas égal au moment de sa quantité de mouvement totale.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink noprint"><a href="#cite_ref-2">↑</a> </span><span class="reference-text"><span class="ouvrage" id="KaneSternheim1997"><span class="ouvrage" id="Joseph_KaneMorton_Sternheim1997">Joseph <span class="nom_auteur">Kane</span> et Morton <span class="nom_auteur">Sternheim</span>, <cite class="italique">Physique: plus de 1900 problèmes et exercices, plus de 800 solutions</cite>, Masson, <abbr class="abbr" title="collection">coll.</abbr> « Enseignement de la physique », <time>1997</time> <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-2-225-83137-9" title="Spécial:Ouvrages de référence/978-2-225-83137-9"><span class="nowrap">978-2-225-83137-9</span></a>)</small>, <abbr class="abbr" title="page">p.</abbr> 172<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physique%3A+plus+de+1900+probl%C3%A8mes+et+exercices%2C+plus+de+800+solutions&rft.pub=Masson&rft.aulast=Kane&rft.aufirst=Joseph&rft.au=Sternheim%2C+Morton&rft.date=1997&rft.pages=172&rft.isbn=978-2-225-83137-9&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AMoment+cin%C3%A9tique"></span></span></span>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink noprint"><a href="#cite_ref-3">↑</a> </span><span class="reference-text"><span class="ouvrage">« <a rel="nofollow" class="external text" href="https://www.futura-sciences.com/sciences/definitions/physique-moment-cinetique-9264/"><cite style="font-style:normal;">Moment cinétique : qu'est-ce que c'est ?</cite></a> », sur <span class="italique">Futura</span> <small style="line-height:1em;">(consulté le <time class="nowrap" datetime="2023-12-12" data-sort-value="2023-12-12">12 décembre 2023</time>)</small></span>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink noprint"><a href="#cite_ref-4">↑</a> </span><span class="reference-text">La notation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"></span> est celle du <a href="/wiki/Produit_vectoriel" title="Produit vectoriel">produit vectoriel</a>.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink noprint"><a href="#cite_ref-5">↑</a> </span><span class="reference-text">Par rapport à un point O mobile dans (<i>R</i>), le théorème du moment cinétique s'écrit : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} {\overrightarrow {L_{\mathrm {O} }}}}{\mathrm {d} t}}+{\overrightarrow {v_{\mathrm {O} }}}\wedge {\vec {p}}=\sum _{i}{\overrightarrow {{\mathcal {M}}_{\mathrm {O} }}}\left({\overrightarrow {F_{i}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} {\overrightarrow {L_{\mathrm {O} }}}}{\mathrm {d} t}}+{\overrightarrow {v_{\mathrm {O} }}}\wedge {\vec {p}}=\sum _{i}{\overrightarrow {{\mathcal {M}}_{\mathrm {O} }}}\left({\overrightarrow {F_{i}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33707222169ada6f90099c450d3c57fbff4b036c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-top: -0.344ex; width:32.273ex; height:8.009ex;" alt="{\displaystyle {\frac {\mathrm {d} {\overrightarrow {L_{\mathrm {O} }}}}{\mathrm {d} t}}+{\overrightarrow {v_{\mathrm {O} }}}\wedge {\vec {p}}=\sum _{i}{\overrightarrow {{\mathcal {M}}_{\mathrm {O} }}}\left({\overrightarrow {F_{i}}}\right)}"></span>, la seule différence vient de l'addition d'un terme complémentaire <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {v_{\mathrm {O} }}}\wedge {\vec {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {v_{\mathrm {O} }}}\wedge {\vec {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75ff97dac62ffcc7ab00321b11bf46b471aa682b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.34ex; width:6.676ex; height:3.509ex;" alt="{\displaystyle {\overrightarrow {v_{\mathrm {O} }}}\wedge {\vec {p}}}"></span> dans le membre de gauche de la relation précédente.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink noprint"><a href="#cite_ref-6">↑</a> </span><span class="reference-text">De façon plus rigoureuse, il faut considérer un axe Δ<sub>O</sub> passant par O et perpendiculaire au plan formé par <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {F}},{\vec {r}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {F}},{\vec {r}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6596d0ba66a67eba5ed69147f910e6a2830978d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.837ex; height:3.343ex;" alt="{\displaystyle ({\vec {F}},{\vec {r}})}"></span>, de vecteur unitaire <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {e_{\Delta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {e_{\Delta }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e6c2e952838cc930856bcd0b48f411a041f8973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.34ex; width:2.815ex; height:3.509ex;" alt="{\displaystyle {\overrightarrow {e_{\Delta }}}}"></span>. Le moment par rapport à l'axe Δ<sub>O</sub> est alors donné par <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}_{\Delta _{\mathrm {O} }}={\overrightarrow {e_{\Delta }}}\cdot \left({\vec {r}}\wedge {\vec {F}}\right)={\vec {F}}\cdot \left({\overrightarrow {e}}_{\Delta }\wedge {\vec {r}}\right)=r{\vec {F}}\cdot \left({\overrightarrow {e}}_{\Delta }\wedge {\vec {r}}_{r}\right)=(r\cos \theta )F=d_{\Delta }F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo>→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> </mrow> </msub> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo>→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> </mrow> </msub> <mo>∧</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mi>F</mi> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> </mrow> </msub> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}_{\Delta _{\mathrm {O} }}={\overrightarrow {e_{\Delta }}}\cdot \left({\vec {r}}\wedge {\vec {F}}\right)={\vec {F}}\cdot \left({\overrightarrow {e}}_{\Delta }\wedge {\vec {r}}\right)=r{\vec {F}}\cdot \left({\overrightarrow {e}}_{\Delta }\wedge {\vec {r}}_{r}\right)=(r\cos \theta )F=d_{\Delta }F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef1ca0046a5d301ad06da4c95bd75c83b4c22c73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:78.25ex; height:4.843ex;" alt="{\displaystyle {\mathcal {M}}_{\Delta _{\mathrm {O} }}={\overrightarrow {e_{\Delta }}}\cdot \left({\vec {r}}\wedge {\vec {F}}\right)={\vec {F}}\cdot \left({\overrightarrow {e}}_{\Delta }\wedge {\vec {r}}\right)=r{\vec {F}}\cdot \left({\overrightarrow {e}}_{\Delta }\wedge {\vec {r}}_{r}\right)=(r\cos \theta )F=d_{\Delta }F}"></span>, avec <i>θ</i> angle entre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aec3c9ce13b53e9e24c98e7cce4212627884c91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef40edff397a115ecdce7d3518001dfcc7f37d9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.771ex; height:2.843ex;" alt="{\displaystyle {\vec {F}}}"></span>, <i>d</i><sub>Δ</sub> étant la distance entre l'axe Δ<sub>O</sub> et la droite support de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef40edff397a115ecdce7d3518001dfcc7f37d9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.771ex; height:2.843ex;" alt="{\displaystyle {\vec {F}}}"></span> (ou bras de levier). Cette quantité traduit bien l'effet de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef40edff397a115ecdce7d3518001dfcc7f37d9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.771ex; height:2.843ex;" alt="{\displaystyle {\vec {F}}}"></span> en ce qui concerne la rotation de M autour de l'axe Δ<sub>O</sub>.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink noprint"><a href="#cite_ref-7">↑</a> </span><span class="reference-text">De façon simple, ce vecteur traduit la rotation d'un système de coordonnées rigidement lié au solide par rapport à (<i>R</i>). Il s'agit aussi du vecteur rotation propre par rapport à (<i>R</i><sup>*</sup>) puisque par définition ce référentiel est en translation relativement à (<i>R</i>).</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink noprint"><a href="#cite_ref-8">↑</a> </span><span class="reference-text">Cette relation est directement liée au fait que la distance entre deux points matériels quelconques est supposée invariante.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink noprint"><a href="#cite_ref-9">↑</a> </span><span class="reference-text">La démonstration se fait de la même façon en formalisme Lagrangien, en tenant compte du fait que par définition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}={\frac {\partial L}{\partial {\dot {q_{i}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}={\frac {\partial L}{\partial {\dot {q_{i}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e2e21f8e5f34bdc290dbe3571d462402b429230" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.089ex; width:9.148ex; height:5.843ex;" alt="{\displaystyle p_{i}={\frac {\partial L}{\partial {\dot {q_{i}}}}}}"></span>.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink noprint"><a href="#cite_ref-10">↑</a> </span><span class="reference-text">En présence d'un champ électromagnétique, et en coordonnées cartésiennes, le moment conjugué est pour une particule de charge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2752dcbff884354069fe332b8e51eb0a70a531b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.837ex; height:2.009ex;" alt="{\displaystyle q_{i}}"></span> est donné par <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p_{i}}}=m_{i}{\vec {v_{i}}}+q_{i}{\vec {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p_{i}}}=m_{i}{\vec {v_{i}}}+q_{i}{\vec {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a19f388fbb669c144cf784c8842d85dce3db5e71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.006ex; height:3.343ex;" alt="{\displaystyle {\vec {p_{i}}}=m_{i}{\vec {v_{i}}}+q_{i}{\vec {A}}}"></span>.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink noprint"><a href="#cite_ref-11">↑</a> </span><span class="reference-text">Dans cette situation cependant s'ajoute le poids de la masselotte attachée au fils, qui est une force non-centrale</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink noprint"><a href="#cite_ref-12">↑</a> </span><span class="reference-text">Cependant l'exemple le plus important (et « historique ») de mouvement à force centrale est le <a href="/wiki/Probl%C3%A8me_%C3%A0_deux_corps" title="Problème à deux corps">problème à deux corps</a>, pour lequel on considère deux points matériels en interaction gravitationnelle. Comme il est indiqué dans l'article qui y est consacré, il se ramène en fait à un problème à un seul corps (particule fictive) soumis à une force centrale, aussi la situation envisagée ici à un intérêt.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink noprint"><a href="#cite_ref-13">↑</a> </span><span class="reference-text">Kepler a mis en évidence cette propriété par le calcul, sans pouvoir l'expliquer. C'est Newton en 1687 qui expliquera l'origine de cette « loi ».</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink noprint"><a href="#cite_ref-14">↑</a> </span><span class="reference-text">Cela provient de l'existence, pour ces potentiels, d'une intégrale première additionnelle (pour le potentiel coulombien, il s'agit du <a href="/wiki/Vecteur_de_Runge-Lenz" title="Vecteur de Runge-Lenz">vecteur de Runge-Lenz</a>), associé à une <a href="/wiki/Sym%C3%A9trie" title="Symétrie">symétrie</a> supplémentaire (par transformation du groupe <i>O(4)</i>).</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink noprint"><a href="#cite_ref-15">↑</a> </span><span class="reference-text">Pour cette partie, cf. <span class="ouvrage" id="GoldsteinPoole_Jr.Safko"><span class="ouvrage" id="Herbert_GoldsteinCharles_P._Poole_Jr.John_L._Safko">Herbert Goldstein, Charles P. Poole Jr. et John L. Safko, <cite class="italique"><span class="lang-en" lang="en">Classical Mechanics</span></cite> <small>[<a href="/wiki/R%C3%A9f%C3%A9rence:Classical_mechanics_(Goldstein)" title="Référence:Classical mechanics (Goldstein)">détail des éditions</a>]</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Mechanics&rft.aulast=Goldstein&rft.aufirst=Herbert&rft.au=Charles+P.+Poole+Jr.&rft.au=John+L.+Safko&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AMoment+cin%C3%A9tique"></span></span></span> et <span class="ouvrage" id="LandauLifchits"><span class="ouvrage" id="Lev_LandauEvgueni_Lifchits"><a href="/wiki/Lev_Landau" title="Lev Landau">Lev Landau</a> et <a href="/wiki/Evgueni_Lifchits" title="Evgueni Lifchits">Evgueni Lifchits</a>, <cite class="italique">Physique théorique</cite>, <abbr class="abbr" title="tome">t.</abbr> 2 : <i>Théorie des champs</i> <small>[<a href="/wiki/R%C3%A9f%C3%A9rence:Physique_th%C3%A9orique_(Landau_et_Lifchitz)" title="Référence:Physique théorique (Landau et Lifchitz)">détail des éditions</a>]</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physique+th%C3%A9orique&rft.aulast=Landau&rft.aufirst=Lev&rft.au=Evgueni+Lifchits&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AMoment+cin%C3%A9tique"></span></span></span>, chapitre 2.</span> </li> </ol></div> </div> <div class="mw-heading mw-heading2"><h2 id="Voir_aussi">Voir aussi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moment_cin%C3%A9tique&veaction=edit&section=19" title="Modifier la section : Voir aussi" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Moment_cin%C3%A9tique&action=edit&section=19" title="Modifier le code source de la section : Voir aussi"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r194021218">.mw-parser-output .autres-projets>.titre{text-align:center;margin:0.2em 0}.mw-parser-output .autres-projets>ul{margin:0;padding:0}.mw-parser-output .autres-projets>ul>li{list-style:none;margin:0.2em 0;text-indent:0;padding-left:24px;min-height:20px;text-align:left;display:block}.mw-parser-output .autres-projets>ul>li>a{font-style:italic}@media(max-width:720px){.mw-parser-output .autres-projets{float:none}}</style><div class="autres-projets boite-grise boite-a-droite noprint js-interprojets"> <p class="titre">Sur les autres projets Wikimedia :</p> <ul class="noarchive plainlinks"> <li class="commons"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Angular_momentum?uselang=fr">Moment cinétique</a>, sur <span class="project">Wikimedia Commons</span></li> </ul> </div> <div class="mw-heading mw-heading3"><h3 id="Articles_connexes">Articles connexes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moment_cin%C3%A9tique&veaction=edit&section=20" title="Modifier la section : Articles connexes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Moment_cin%C3%A9tique&action=edit&section=20" title="Modifier le code source de la section : Articles connexes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Moment_(m%C3%A9canique)" class="mw-redirect" title="Moment (mécanique)">Moment (mécanique)</a></li> <li><a href="/wiki/Probl%C3%A8me_%C3%A0_deux_corps" title="Problème à deux corps">Problème à deux corps</a></li> <li><a href="/wiki/Probl%C3%A8me_%C3%A0_N_corps" title="Problème à N corps">Problème à N corps</a></li> <li><a href="/wiki/Moment_cin%C3%A9tique_(m%C3%A9canique_quantique)" title="Moment cinétique (mécanique quantique)">Moment cinétique (mécanique quantique)</a></li> <li><a href="/wiki/Moment_cin%C3%A9tique_(relativit%C3%A9)" title="Moment cinétique (relativité)">Moment cinétique (relativité)</a></li> <li><a href="/wiki/Moment_cin%C3%A9tique_orbital" title="Moment cinétique orbital">Moment cinétique orbital</a> et <a href="/wiki/Spin" title="Spin">spin</a> en mécanique quantique</li> <li><a href="/wiki/Moment_cin%C3%A9tique_sp%C3%A9cifique" title="Moment cinétique spécifique">Moment cinétique spécifique</a></li> <li><a href="/wiki/Moment_d%27inertie" title="Moment d'inertie">Moment d'inertie</a></li> <li><a href="/wiki/Pseudovecteur" title="Pseudovecteur">Pseudovecteur</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Bibliographie">Bibliographie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moment_cin%C3%A9tique&veaction=edit&section=21" title="Modifier la section : Bibliographie" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Moment_cin%C3%A9tique&action=edit&section=21" title="Modifier le code source de la section : Bibliographie"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="ouvrage" id="LandauLifchits"><span class="ouvrage" id="Lev_LandauEvgueni_Lifchits"><a href="/wiki/Lev_Landau" title="Lev Landau">Lev Landau</a> et <a href="/wiki/Evgueni_Lifchits" title="Evgueni Lifchits">Evgueni Lifchits</a>, <cite class="italique">Physique théorique</cite>, <abbr class="abbr" title="tome">t.</abbr> 1 : <i>Mécanique</i> <small>[<a href="/wiki/R%C3%A9f%C3%A9rence:Physique_th%C3%A9orique_(Landau_et_Lifchitz)" title="Référence:Physique théorique (Landau et Lifchitz)">détail des éditions</a>]</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physique+th%C3%A9orique&rft.aulast=Landau&rft.aufirst=Lev&rft.au=Evgueni+Lifchits&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AMoment+cin%C3%A9tique"></span></span></span></li> <li><span class="ouvrage" id="GoldsteinPoole_Jr.Safko"><span class="ouvrage" id="Herbert_GoldsteinCharles_P._Poole_Jr.John_L._Safko">Herbert Goldstein, Charles P. Poole Jr. et John L. Safko, <cite class="italique"><span class="lang-en" lang="en">Classical Mechanics</span></cite> <small>[<a href="/wiki/R%C3%A9f%C3%A9rence:Classical_mechanics_(Goldstein)" title="Référence:Classical mechanics (Goldstein)">détail des éditions</a>]</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Mechanics&rft.aulast=Goldstein&rft.aufirst=Herbert&rft.au=Charles+P.+Poole+Jr.&rft.au=John+L.+Safko&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AMoment+cin%C3%A9tique"></span></span></span></li> <li>Perez, <i>Cours de physique : mécanique</i> - <abbr class="abbr" title="Sixième">6<sup>e</sup></abbr> édition, Masson, Paris, 2001.</li> <li>Yo-Yo, billard, boomerang, la physique des objets tournants, ed Belin,2001, <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/2_84245_016_7" title="Spécial:Ouvrages de référence/2 84245 016 7"><span class="nowrap">2 84245 016 7</span></a>)</small></li></ul> <div class="mw-heading mw-heading3"><h3 id="Liens_externes">Liens externes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moment_cin%C3%A9tique&veaction=edit&section=22" title="Modifier la section : Liens externes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Moment_cin%C3%A9tique&action=edit&section=22" title="Modifier le code source de la section : Liens externes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Vidéos d'illustration : <ul><li><abbr class="abbr indicateur-langue" title="Langue : français">(fr)</abbr> <a rel="nofollow" class="external text" href="http://videosphysique.blogspot.fr/2011/02/conservation-du-moment-cinetique-avec.html">Une vidéo explicative sur la conservation du moment cinétique</a></li> <li><abbr class="abbr indicateur-langue" title="Langue : français">(fr)</abbr> <a rel="nofollow" class="external text" href="http://www.canal-u.tv/video/tele2sciences/la_centrifugeuse.6676">Une autre vidéo illustrant la conservation du moment cinétique</a></li> <li><abbr class="abbr indicateur-langue" title="Langue : français">(fr)</abbr> <a rel="nofollow" class="external text" href="http://www.fundp.ac.be/sciences/physique/udp/videos/moment-cinetique.html">Liens vers diverses vidéos illustrant la notion de moment cinétique et sa conservation</a></li></ul></li></ul> <ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer" typeof="mw:File"><a href="/wiki/Portail:Physique" title="Portail de la physique"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circle-icons-physics-logo.svg/24px-Circle-icons-physics-logo.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circle-icons-physics-logo.svg/36px-Circle-icons-physics-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circle-icons-physics-logo.svg/48px-Circle-icons-physics-logo.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:Physique" title="Portail:Physique">Portail de la physique</a></span> </span></li> </ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; 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